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Properties

Label 2-76-19.7-c5-0-3
Degree $2$
Conductor $76$
Sign $-0.291 - 0.956i$
Analytic cond. $12.1891$
Root an. cond. $3.49129$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (10.4 + 18.1i)3-s + (50.5 + 87.6i)5-s + 95.5·7-s + (−97.0 + 168. i)9-s − 119.·11-s + (297. − 515. i)13-s + (−1.05e3 + 1.83e3i)15-s + (−459. − 796. i)17-s + (−806. + 1.35e3i)19-s + (998. + 1.73e3i)21-s + (2.12e3 − 3.67e3i)23-s + (−3.55e3 + 6.16e3i)25-s + 1.02e3·27-s + (−2.21e3 + 3.83e3i)29-s − 4.95e3·31-s + ⋯
L(s)  = 1  + (0.670 + 1.16i)3-s + (0.905 + 1.56i)5-s + 0.736·7-s + (−0.399 + 0.691i)9-s − 0.297·11-s + (0.488 − 0.845i)13-s + (−1.21 + 2.10i)15-s + (−0.385 − 0.668i)17-s + (−0.512 + 0.858i)19-s + (0.494 + 0.856i)21-s + (0.837 − 1.45i)23-s + (−1.13 + 1.97i)25-s + 0.269·27-s + (−0.489 + 0.846i)29-s − 0.926·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.291 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.291 - 0.956i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (45, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.291 - 0.956i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.57907 + 2.13256i\)
\(L(\frac12)\) \(\approx\) \(1.57907 + 2.13256i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (806. - 1.35e3i)T \)
good3 \( 1 + (-10.4 - 18.1i)T + (-121.5 + 210. i)T^{2} \)
5 \( 1 + (-50.5 - 87.6i)T + (-1.56e3 + 2.70e3i)T^{2} \)
7 \( 1 - 95.5T + 1.68e4T^{2} \)
11 \( 1 + 119.T + 1.61e5T^{2} \)
13 \( 1 + (-297. + 515. i)T + (-1.85e5 - 3.21e5i)T^{2} \)
17 \( 1 + (459. + 796. i)T + (-7.09e5 + 1.22e6i)T^{2} \)
23 \( 1 + (-2.12e3 + 3.67e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (2.21e3 - 3.83e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 + 4.95e3T + 2.86e7T^{2} \)
37 \( 1 - 7.65e3T + 6.93e7T^{2} \)
41 \( 1 + (-2.00e3 - 3.47e3i)T + (-5.79e7 + 1.00e8i)T^{2} \)
43 \( 1 + (5.22e3 + 9.05e3i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 + (-4.31e3 + 7.47e3i)T + (-1.14e8 - 1.98e8i)T^{2} \)
53 \( 1 + (-1.85e4 + 3.21e4i)T + (-2.09e8 - 3.62e8i)T^{2} \)
59 \( 1 + (2.07e4 + 3.59e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (2.24e3 - 3.88e3i)T + (-4.22e8 - 7.31e8i)T^{2} \)
67 \( 1 + (-1.84e4 + 3.20e4i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-2.34e4 - 4.05e4i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 + (-4.06e4 - 7.04e4i)T + (-1.03e9 + 1.79e9i)T^{2} \)
79 \( 1 + (-6.65e3 - 1.15e4i)T + (-1.53e9 + 2.66e9i)T^{2} \)
83 \( 1 + 1.83e4T + 3.93e9T^{2} \)
89 \( 1 + (7.30e3 - 1.26e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-6.18e4 - 1.07e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.45902824812964375336405672900, −13.06767168592145616826567699241, −11.00865810652434077317103972541, −10.55595484346080353002016560468, −9.566168612459879358469206339995, −8.288699883512590106616819931104, −6.74789094141811116817961025160, −5.24654812373197270148590856310, −3.55058596530746004500176527844, −2.38620830905702919151804382089, 1.21444795365821826214036194496, 2.05808943599380001968011393394, 4.59547545578381292555230617349, 5.98127381061296160213726924068, 7.55275620132058870258488229836, 8.638669721208146713235788148860, 9.303945713972113141916928413872, 11.20335170558049937414506404362, 12.53483303189991341196990171093, 13.32863588821044803144351727971

Graph of the $Z$-function along the critical line