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Properties

Label 2-616-11.3-c1-0-5
Degree 22
Conductor 616616
Sign −0.147−0.989i-0.147 - 0.989i
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.665 + 2.04i)3-s + (2.20 + 1.60i)5-s + (−0.309 + 0.951i)7-s + (−1.32 + 0.962i)9-s + (2.37 + 2.31i)11-s + (0.966 − 0.702i)13-s + (−1.81 + 5.58i)15-s + (−3.00 − 2.18i)17-s + (−0.701 − 2.15i)19-s − 2.15·21-s + 1.70·23-s + (0.751 + 2.31i)25-s + (2.37 + 1.72i)27-s + (1.38 − 4.25i)29-s + (−5.51 + 4.00i)31-s + ⋯
L(s)  = 1  + (0.384 + 1.18i)3-s + (0.986 + 0.716i)5-s + (−0.116 + 0.359i)7-s + (−0.441 + 0.320i)9-s + (0.717 + 0.696i)11-s + (0.268 − 0.194i)13-s + (−0.468 + 1.44i)15-s + (−0.728 − 0.529i)17-s + (−0.160 − 0.495i)19-s − 0.469·21-s + 0.354·23-s + (0.150 + 0.462i)25-s + (0.456 + 0.332i)27-s + (0.257 − 0.791i)29-s + (−0.990 + 0.719i)31-s + ⋯

Functional equation

Λ(s)=(616s/2 ΓC(s) L(s)=((−0.147−0.989i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(616s/2 ΓC(s+1/2) L(s)=((−0.147−0.989i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    23⋅7⋅112^{3} \cdot 7 \cdot 11
Sign: −0.147−0.989i-0.147 - 0.989i
Analytic conductor: 4.918784.91878
Root analytic conductor: 2.217832.21783
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ616(113,⋅)\chi_{616} (113, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 616, ( :1/2), −0.147−0.989i)(2,\ 616,\ (\ :1/2),\ -0.147 - 0.989i)

Particular Values

L(1)L(1) ≈\approx 1.30217+1.51020i1.30217 + 1.51020i
L(12)L(\frac12) ≈\approx 1.30217+1.51020i1.30217 + 1.51020i
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

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+(0.309−0.951i)T 1 + (0.309 - 0.951i)T
11 1+(−2.37−2.31i)T 1 + (-2.37 - 2.31i)T
good3 1+(−0.665−2.04i)T+(−2.42+1.76i)T2 1 + (-0.665 - 2.04i)T + (-2.42 + 1.76i)T^{2}
5 1+(−2.20−1.60i)T+(1.54+4.75i)T2 1 + (-2.20 - 1.60i)T + (1.54 + 4.75i)T^{2}
13 1+(−0.966+0.702i)T+(4.01−12.3i)T2 1 + (-0.966 + 0.702i)T + (4.01 - 12.3i)T^{2}
17 1+(3.00+2.18i)T+(5.25+16.1i)T2 1 + (3.00 + 2.18i)T + (5.25 + 16.1i)T^{2}
19 1+(0.701+2.15i)T+(−15.3+11.1i)T2 1 + (0.701 + 2.15i)T + (-15.3 + 11.1i)T^{2}
23 1−1.70T+23T2 1 - 1.70T + 23T^{2}
29 1+(−1.38+4.25i)T+(−23.4−17.0i)T2 1 + (-1.38 + 4.25i)T + (-23.4 - 17.0i)T^{2}
31 1+(5.51−4.00i)T+(9.57−29.4i)T2 1 + (5.51 - 4.00i)T + (9.57 - 29.4i)T^{2}
37 1+(−1.79+5.53i)T+(−29.9−21.7i)T2 1 + (-1.79 + 5.53i)T + (-29.9 - 21.7i)T^{2}
41 1+(0.488+1.50i)T+(−33.1+24.0i)T2 1 + (0.488 + 1.50i)T + (-33.1 + 24.0i)T^{2}
43 1+9.77T+43T2 1 + 9.77T + 43T^{2}
47 1+(0.129+0.399i)T+(−38.0+27.6i)T2 1 + (0.129 + 0.399i)T + (-38.0 + 27.6i)T^{2}
53 1+(−1.64+1.19i)T+(16.3−50.4i)T2 1 + (-1.64 + 1.19i)T + (16.3 - 50.4i)T^{2}
59 1+(2.39−7.36i)T+(−47.7−34.6i)T2 1 + (2.39 - 7.36i)T + (-47.7 - 34.6i)T^{2}
61 1+(4.35+3.16i)T+(18.8+58.0i)T2 1 + (4.35 + 3.16i)T + (18.8 + 58.0i)T^{2}
67 1−8.76T+67T2 1 - 8.76T + 67T^{2}
71 1+(1.26+0.918i)T+(21.9+67.5i)T2 1 + (1.26 + 0.918i)T + (21.9 + 67.5i)T^{2}
73 1+(3.96−12.2i)T+(−59.0−42.9i)T2 1 + (3.96 - 12.2i)T + (-59.0 - 42.9i)T^{2}
79 1+(−11.7+8.55i)T+(24.4−75.1i)T2 1 + (-11.7 + 8.55i)T + (24.4 - 75.1i)T^{2}
83 1+(−7.57−5.50i)T+(25.6+78.9i)T2 1 + (-7.57 - 5.50i)T + (25.6 + 78.9i)T^{2}
89 1+4.67T+89T2 1 + 4.67T + 89T^{2}
97 1+(−9.56+6.95i)T+(29.9−92.2i)T2 1 + (-9.56 + 6.95i)T + (29.9 - 92.2i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−10.63310631771328244853608809635, −9.895206103633825619537556286169, −9.316386303744147270749332436522, −8.669441717869679722009520283870, −7.12298918183202790273104590239, −6.38365000892134938788607989660, −5.25370202898430171824920172212, −4.28054051783165983437444087479, −3.17259028702294530491834017962, −2.09407558287129768473336050222, 1.19365066101925706676558967915, 2.01277474078249752843598137209, 3.55040682242966118322683275179, 4.93062608999479818856335364843, 6.17553368541704022478902061621, 6.63913476690318699595997519206, 7.80451680752480914677197293340, 8.656288320573451666801136682192, 9.272371604051189424712416242173, 10.33906043171952564823802785772

Graph of the ZZ-function along the critical line