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Properties

Label 2-76-19.11-c5-0-2
Degree $2$
Conductor $76$
Sign $0.292 - 0.956i$
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  + (−3.30 + 5.72i)3-s + (12.6 − 21.8i)5-s − 40.7·7-s + (99.6 + 172. i)9-s + 324.·11-s + (−48.8 − 84.6i)13-s + (83.3 + 144. i)15-s + (−1.05e3 + 1.83e3i)17-s + (806. + 1.35e3i)19-s + (134. − 232. i)21-s + (1.50e3 + 2.60e3i)23-s + (1.24e3 + 2.15e3i)25-s − 2.92e3·27-s + (−527. − 914. i)29-s + 7.60e3·31-s + ⋯
L(s)  = 1  + (−0.211 + 0.367i)3-s + (0.225 − 0.390i)5-s − 0.314·7-s + (0.410 + 0.710i)9-s + 0.809·11-s + (−0.0801 − 0.138i)13-s + (0.0955 + 0.165i)15-s + (−0.887 + 1.53i)17-s + (0.512 + 0.858i)19-s + (0.0665 − 0.115i)21-s + (0.593 + 1.02i)23-s + (0.398 + 0.689i)25-s − 0.771·27-s + (−0.116 − 0.201i)29-s + 1.42·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.292 - 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.292 - 0.956i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ 0.292 - 0.956i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.26519 + 0.936500i\)
\(L(\frac12)\) \(\approx\) \(1.26519 + 0.936500i\)
\(L(\frac{7}{2})\) 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 \( 1 + (3.30 - 5.72i)T + (-121.5 - 210. i)T^{2} \)
5 \( 1 + (-12.6 + 21.8i)T + (-1.56e3 - 2.70e3i)T^{2} \)
7 \( 1 + 40.7T + 1.68e4T^{2} \)
11 \( 1 - 324.T + 1.61e5T^{2} \)
13 \( 1 + (48.8 + 84.6i)T + (-1.85e5 + 3.21e5i)T^{2} \)
17 \( 1 + (1.05e3 - 1.83e3i)T + (-7.09e5 - 1.22e6i)T^{2} \)
23 \( 1 + (-1.50e3 - 2.60e3i)T + (-3.21e6 + 5.57e6i)T^{2} \)
29 \( 1 + (527. + 914. i)T + (-1.02e7 + 1.77e7i)T^{2} \)
31 \( 1 - 7.60e3T + 2.86e7T^{2} \)
37 \( 1 + 2.05e3T + 6.93e7T^{2} \)
41 \( 1 + (-230. + 399. i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (188. - 326. i)T + (-7.35e7 - 1.27e8i)T^{2} \)
47 \( 1 + (8.35e3 + 1.44e4i)T + (-1.14e8 + 1.98e8i)T^{2} \)
53 \( 1 + (-5.23e3 - 9.06e3i)T + (-2.09e8 + 3.62e8i)T^{2} \)
59 \( 1 + (-2.37e4 + 4.11e4i)T + (-3.57e8 - 6.19e8i)T^{2} \)
61 \( 1 + (1.37e4 + 2.37e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (-1.72e4 - 2.98e4i)T + (-6.75e8 + 1.16e9i)T^{2} \)
71 \( 1 + (-1.87e3 + 3.25e3i)T + (-9.02e8 - 1.56e9i)T^{2} \)
73 \( 1 + (-6.14e3 + 1.06e4i)T + (-1.03e9 - 1.79e9i)T^{2} \)
79 \( 1 + (-2.67e4 + 4.62e4i)T + (-1.53e9 - 2.66e9i)T^{2} \)
83 \( 1 + 1.31e3T + 3.93e9T^{2} \)
89 \( 1 + (-5.17e4 - 8.96e4i)T + (-2.79e9 + 4.83e9i)T^{2} \)
97 \( 1 + (5.29e4 - 9.17e4i)T + (-4.29e9 - 7.43e9i)T^{2} \)
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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

−13.59060229399234112163920049816, −12.77347101374574785979503413894, −11.48495693606202570077227691336, −10.35137101131407048247510805352, −9.356681431625113854563005873274, −8.066134429345203140184228216724, −6.52950495428139185391074014616, −5.16920343325840858677878731384, −3.78742955095623954082593954240, −1.59010436445537716865881138340, 0.75925068563071204718040967181, 2.79347230476267010047870393896, 4.59590716567054839306524384247, 6.47169425606581479928830964777, 7.02112402548254064014900635029, 8.905684512814206904952883264924, 9.833576397711149393417724077680, 11.27276521929103709066403513307, 12.16046131275554223164671336246, 13.31562786328110373005360453382

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