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Properties

Label 1-14e3-2744.501-r0-0-0
Degree 11
Conductor 27442744
Sign −0.778−0.628i-0.778 - 0.628i
Analytic cond. 12.743012.7430
Root an. cond. 12.743012.7430
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.589 + 0.807i)3-s + (0.536 + 0.843i)5-s + (−0.304 − 0.952i)9-s + (−0.687 + 0.725i)11-s + (0.572 − 0.820i)13-s + (−0.997 − 0.0640i)15-s + (−0.0534 + 0.998i)17-s + (0.5 − 0.866i)19-s + (−0.961 + 0.274i)23-s + (−0.424 + 0.905i)25-s + (0.949 + 0.315i)27-s + (−0.718 + 0.695i)29-s + (0.826 + 0.563i)31-s + (−0.180 − 0.983i)33-s + (−0.891 + 0.453i)37-s + ⋯
L(s)  = 1  + (−0.589 + 0.807i)3-s + (0.536 + 0.843i)5-s + (−0.304 − 0.952i)9-s + (−0.687 + 0.725i)11-s + (0.572 − 0.820i)13-s + (−0.997 − 0.0640i)15-s + (−0.0534 + 0.998i)17-s + (0.5 − 0.866i)19-s + (−0.961 + 0.274i)23-s + (−0.424 + 0.905i)25-s + (0.949 + 0.315i)27-s + (−0.718 + 0.695i)29-s + (0.826 + 0.563i)31-s + (−0.180 − 0.983i)33-s + (−0.891 + 0.453i)37-s + ⋯

Functional equation

Λ(s)=(2744s/2 ΓR(s) L(s)=((−0.778−0.628i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(2744s/2 ΓR(s) L(s)=((−0.778−0.628i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 2744 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.778 - 0.628i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 27442744    =    23⋅732^{3} \cdot 7^{3}
Sign: −0.778−0.628i-0.778 - 0.628i
Analytic conductor: 12.743012.7430
Root analytic conductor: 12.743012.7430
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ2744(501,⋅)\chi_{2744} (501, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 2744, (0: ), −0.778−0.628i)(1,\ 2744,\ (0:\ ),\ -0.778 - 0.628i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx −0.1508219990+0.4268397973i-0.1508219990 + 0.4268397973i
L(12)L(\frac12) ≈\approx −0.1508219990+0.4268397973i-0.1508219990 + 0.4268397973i
L(1)L(1) ≈\approx 0.6946805950+0.3660855761i0.6946805950 + 0.3660855761i
L(1)L(1) ≈\approx 0.6946805950+0.3660855761i0.6946805950 + 0.3660855761i

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
7 1 1
good3 1+(−0.589+0.807i)T 1 + (-0.589 + 0.807i)T
5 1+(0.536+0.843i)T 1 + (0.536 + 0.843i)T
11 1+(−0.687+0.725i)T 1 + (-0.687 + 0.725i)T
13 1+(0.572−0.820i)T 1 + (0.572 - 0.820i)T
17 1+(−0.0534+0.998i)T 1 + (-0.0534 + 0.998i)T
19 1+(0.5−0.866i)T 1 + (0.5 - 0.866i)T
23 1+(−0.961+0.274i)T 1 + (-0.961 + 0.274i)T
29 1+(−0.718+0.695i)T 1 + (-0.718 + 0.695i)T
31 1+(0.826+0.563i)T 1 + (0.826 + 0.563i)T
37 1+(−0.891+0.453i)T 1 + (-0.891 + 0.453i)T
41 1+(−0.761−0.648i)T 1 + (-0.761 - 0.648i)T
43 1+(−0.801−0.598i)T 1 + (-0.801 - 0.598i)T
47 1+(0.443−0.896i)T 1 + (0.443 - 0.896i)T
53 1+(0.999+0.0213i)T 1 + (0.999 + 0.0213i)T
59 1+(−0.942−0.335i)T 1 + (-0.942 - 0.335i)T
61 1+(0.640+0.768i)T 1 + (0.640 + 0.768i)T
67 1+(−0.0747−0.997i)T 1 + (-0.0747 - 0.997i)T
71 1+(0.718+0.695i)T 1 + (0.718 + 0.695i)T
73 1+(0.443+0.896i)T 1 + (0.443 + 0.896i)T
79 1+(−0.733−0.680i)T 1 + (-0.733 - 0.680i)T
83 1+(−0.926+0.375i)T 1 + (-0.926 + 0.375i)T
89 1+(−0.860+0.509i)T 1 + (-0.860 + 0.509i)T
97 1+(−0.900+0.433i)T 1 + (-0.900 + 0.433i)T
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   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−18.51607714023535254169937612318, −18.33730015584345583351286093731, −17.42407346942339352933637789697, −16.60493895882876286720892579048, −16.326673377519419081657854597618, −15.580338927345531212085160242764, −14.11327421696849162887354442185, −13.768314202471742505889307185631, −13.229616393279811291200771349, −12.3725470843140062432504088963, −11.75518056911955297578896182089, −11.16863978617358685680292144953, −10.155972935202792695644019245317, −9.49684147824590401346843396893, −8.44884265066158506623897640483, −8.04185780442628059892966016314, −7.09065522684483653077418574514, −6.13736085311417955690456363307, −5.72674879309378479705352508048, −4.95125889763282811262874292024, −4.08297775828639929673056778800, −2.825440004839329235559656943921, −1.92910031812000320719491858964, −1.20806910060120664506873074843, −0.154451569519394403536667882759, 1.4186021208928518518191912727, 2.47233480803009648322422814445, 3.363676534747922204105590188482, 3.98645530027683077857547671117, 5.27005783663004397400371516221, 5.45364109277252047218929079583, 6.51186181313969195814904803920, 7.067890507029383307082398925248, 8.153733971746921454309033748480, 8.97300171331925594374308303461, 9.994366597313163484652123200510, 10.30745572945852970612626440057, 10.85760781029075498756276663407, 11.71591940972765743116657211424, 12.49177380463342284877315501286, 13.37712085549127309237711745914, 14.02425801067938940535306420333, 15.19087251988365778549812723972, 15.23363300888199017615768988207, 16.000734707574596374321120930678, 17.07747993764854829524616359336, 17.54360281420470128192785916013, 18.14341002566927635180787056000, 18.694806411022703805613348007249, 19.94570736601637030943637411357

Graph of the ZZ-function along the critical line