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Properties

Label 1-14e3-2744.501-r0-0-0
Degree $1$
Conductor $2744$
Sign $-0.778 - 0.628i$
Analytic cond. $12.7430$
Root an. cond. $12.7430$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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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

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

Invariants

Degree: \(1\)
Conductor: \(2744\)    =    \(2^{3} \cdot 7^{3}\)
Sign: $-0.778 - 0.628i$
Analytic conductor: \(12.7430\)
Root analytic conductor: \(12.7430\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2744} (501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2744,\ (0:\ ),\ -0.778 - 0.628i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1508219990 + 0.4268397973i\)
\(L(\frac12)\) \(\approx\) \(-0.1508219990 + 0.4268397973i\)
\(L(1)\) \(\approx\) \(0.6946805950 + 0.3660855761i\)
\(L(1)\) \(\approx\) \(0.6946805950 + 0.3660855761i\)

Euler product

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

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 $Z$-function along the critical line