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Properties

Label 2-76-76.75-c5-0-9
Degree $2$
Conductor $76$
Sign $-0.908 + 0.418i$
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  + (0.877 + 5.58i)2-s − 10.0·3-s + (−30.4 + 9.80i)4-s + 80.9·5-s + (−8.82 − 56.2i)6-s + 200. i·7-s + (−81.4 − 161. i)8-s − 141.·9-s + (70.9 + 452. i)10-s + 464. i·11-s + (306. − 98.6i)12-s − 659. i·13-s + (−1.11e3 + 175. i)14-s − 814.·15-s + (831. − 597. i)16-s − 1.15e3·17-s + ⋯
L(s)  = 1  + (0.155 + 0.987i)2-s − 0.645·3-s + (−0.951 + 0.306i)4-s + 1.44·5-s + (−0.100 − 0.638i)6-s + 1.54i·7-s + (−0.450 − 0.892i)8-s − 0.582·9-s + (0.224 + 1.42i)10-s + 1.15i·11-s + (0.614 − 0.197i)12-s − 1.08i·13-s + (−1.52 + 0.239i)14-s − 0.934·15-s + (0.812 − 0.583i)16-s − 0.971·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.908 + 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.908 + 0.418i$
Analytic conductor: \(12.1891\)
Root analytic conductor: \(3.49129\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :5/2),\ -0.908 + 0.418i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.203414 - 0.928003i\)
\(L(\frac12)\) \(\approx\) \(0.203414 - 0.928003i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.877 - 5.58i)T \)
19 \( 1 + (1.56e3 - 188. i)T \)
good3 \( 1 + 10.0T + 243T^{2} \)
5 \( 1 - 80.9T + 3.12e3T^{2} \)
7 \( 1 - 200. iT - 1.68e4T^{2} \)
11 \( 1 - 464. iT - 1.61e5T^{2} \)
13 \( 1 + 659. iT - 3.71e5T^{2} \)
17 \( 1 + 1.15e3T + 1.41e6T^{2} \)
23 \( 1 + 2.53e3iT - 6.43e6T^{2} \)
29 \( 1 - 6.62e3iT - 2.05e7T^{2} \)
31 \( 1 + 1.00e4T + 2.86e7T^{2} \)
37 \( 1 + 2.10e3iT - 6.93e7T^{2} \)
41 \( 1 - 4.12e3iT - 1.15e8T^{2} \)
43 \( 1 - 575. iT - 1.47e8T^{2} \)
47 \( 1 - 1.34e4iT - 2.29e8T^{2} \)
53 \( 1 - 1.83e4iT - 4.18e8T^{2} \)
59 \( 1 - 3.85e4T + 7.14e8T^{2} \)
61 \( 1 - 2.50e4T + 8.44e8T^{2} \)
67 \( 1 - 3.21e3T + 1.35e9T^{2} \)
71 \( 1 + 1.08e4T + 1.80e9T^{2} \)
73 \( 1 - 8.06e4T + 2.07e9T^{2} \)
79 \( 1 + 2.18e4T + 3.07e9T^{2} \)
83 \( 1 - 3.29e4iT - 3.93e9T^{2} \)
89 \( 1 + 3.23e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.55e5iT - 8.58e9T^{2} \)
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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

−14.48744721771731107073171743110, −12.87868375376479219180397684760, −12.51501304154964116980600346076, −10.67449271647962917941399256649, −9.356173032569756091433407962353, −8.568913858579721823037512760752, −6.65564447061341690044025530886, −5.74580053163026392486763238589, −5.06878547844276709062496656173, −2.37580553467912408526039705305, 0.40364931050960452338879274203, 1.98876611898190773804812664244, 3.92809691340301802168421556982, 5.44958166270176897069188879316, 6.54846852221184768139885858482, 8.732204149866140767172506269350, 9.839740151839243736691601753295, 10.88709507020726513290460640120, 11.41123805256265546032762332350, 13.19473865264924725269263302066

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