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Properties

Label 2-76-76.75-c1-0-5
Degree 22
Conductor 7676
Sign 0.603+0.797i0.603 + 0.797i
Analytic cond. 0.6068630.606863
Root an. cond. 0.7790140.779014
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  + (1.06 − 0.927i)2-s − 1.19·3-s + (0.280 − 1.98i)4-s + 1.56·5-s + (−1.28 + 1.11i)6-s + 0.868i·7-s + (−1.53 − 2.37i)8-s − 1.56·9-s + (1.66 − 1.44i)10-s + 3.09i·11-s + (−0.336 + 2.37i)12-s + 4.74i·13-s + (0.804 + 0.927i)14-s − 1.87·15-s + (−3.84 − 1.11i)16-s − 17-s + ⋯
L(s)  = 1  + (0.755 − 0.655i)2-s − 0.692·3-s + (0.140 − 0.990i)4-s + 0.698·5-s + (−0.522 + 0.453i)6-s + 0.328i·7-s + (−0.543 − 0.839i)8-s − 0.520·9-s + (0.527 − 0.457i)10-s + 0.932i·11-s + (−0.0972 + 0.685i)12-s + 1.31i·13-s + (0.215 + 0.247i)14-s − 0.483·15-s + (−0.960 − 0.277i)16-s − 0.242·17-s + ⋯

Functional equation

Λ(s)=(76s/2 ΓC(s) L(s)=((0.603+0.797i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(76s/2 ΓC(s+1/2) L(s)=((0.603+0.797i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 7676    =    22⋅192^{2} \cdot 19
Sign: 0.603+0.797i0.603 + 0.797i
Analytic conductor: 0.6068630.606863
Root analytic conductor: 0.7790140.779014
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ76(75,⋅)\chi_{76} (75, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 76, ( :1/2), 0.603+0.797i)(2,\ 76,\ (\ :1/2),\ 0.603 + 0.797i)

Particular Values

L(1)L(1) ≈\approx 1.03052−0.512508i1.03052 - 0.512508i
L(12)L(\frac12) ≈\approx 1.03052−0.512508i1.03052 - 0.512508i
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+(−1.06+0.927i)T 1 + (-1.06 + 0.927i)T
19 1+(−3.07+3.09i)T 1 + (-3.07 + 3.09i)T
good3 1+1.19T+3T2 1 + 1.19T + 3T^{2}
5 1−1.56T+5T2 1 - 1.56T + 5T^{2}
7 1−0.868iT−7T2 1 - 0.868iT - 7T^{2}
11 1−3.09iT−11T2 1 - 3.09iT - 11T^{2}
13 1−4.74iT−13T2 1 - 4.74iT - 13T^{2}
17 1+T+17T2 1 + T + 17T^{2}
23 1+3.96iT−23T2 1 + 3.96iT - 23T^{2}
29 1+8.45iT−29T2 1 + 8.45iT - 29T^{2}
31 1+4.27T+31T2 1 + 4.27T + 31T^{2}
37 1−3.70iT−37T2 1 - 3.70iT - 37T^{2}
41 1+3.70iT−41T2 1 + 3.70iT - 41T^{2}
43 1−11.0iT−43T2 1 - 11.0iT - 43T^{2}
47 1+9.27iT−47T2 1 + 9.27iT - 47T^{2}
53 1−1.04iT−53T2 1 - 1.04iT - 53T^{2}
59 1−11.6T+59T2 1 - 11.6T + 59T^{2}
61 1+0.684T+61T2 1 + 0.684T + 61T^{2}
67 1+9.74T+67T2 1 + 9.74T + 67T^{2}
71 1+10.9T+71T2 1 + 10.9T + 71T^{2}
73 1−8.12T+73T2 1 - 8.12T + 73T^{2}
79 1+8.01T+79T2 1 + 8.01T + 79T^{2}
83 1−9.65iT−83T2 1 - 9.65iT - 83T^{2}
89 1+5.79iT−89T2 1 + 5.79iT - 89T^{2}
97 1−16.9iT−97T2 1 - 16.9iT - 97T^{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

−14.15056735641180988280449719782, −13.24916265150195017301943112562, −11.98793472986176145255184472531, −11.42202548566248900053695846140, −10.09689573793728439140077686579, −9.132640273946707744970785067147, −6.76656881913769197688988453338, −5.71508598516158499356213175893, −4.50781930093701196262537216478, −2.27673060046378987766924326058, 3.32472764545595268802635285011, 5.42207377499383622179197897989, 5.88743290187322996826442695187, 7.44800365659175530128906231750, 8.784772504140371997895987787647, 10.50749616570065624228243596259, 11.54757108419020966380362268511, 12.72411917044511659573645710998, 13.69803480950288826711553840107, 14.49195505430138544548757172363

Graph of the ZZ-function along the critical line