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Properties

Label 2-76-76.75-c5-0-34
Degree $2$
Conductor $76$
Sign $-0.607 + 0.794i$
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  + (−1.28 − 5.50i)2-s + 16.4·3-s + (−28.6 + 14.1i)4-s + 16.6·5-s + (−21.1 − 90.5i)6-s − 54.8i·7-s + (114. + 139. i)8-s + 27.3·9-s + (−21.4 − 91.8i)10-s − 545. i·11-s + (−471. + 232. i)12-s − 885. i·13-s + (−302. + 70.5i)14-s + 274.·15-s + (623. − 812. i)16-s + 2.30e3·17-s + ⋯
L(s)  = 1  + (−0.227 − 0.973i)2-s + 1.05·3-s + (−0.896 + 0.442i)4-s + 0.298·5-s + (−0.239 − 1.02i)6-s − 0.423i·7-s + (0.634 + 0.772i)8-s + 0.112·9-s + (−0.0677 − 0.290i)10-s − 1.35i·11-s + (−0.945 + 0.466i)12-s − 1.45i·13-s + (−0.412 + 0.0961i)14-s + 0.314·15-s + (0.608 − 0.793i)16-s + 1.93·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.607 + 0.794i$
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: \((2,\ 76,\ (\ :5/2),\ -0.607 + 0.794i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.854824 - 1.73105i\)
\(L(\frac12)\) \(\approx\) \(0.854824 - 1.73105i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.28 + 5.50i)T \)
19 \( 1 + (1.41e3 - 697. i)T \)
good3 \( 1 - 16.4T + 243T^{2} \)
5 \( 1 - 16.6T + 3.12e3T^{2} \)
7 \( 1 + 54.8iT - 1.68e4T^{2} \)
11 \( 1 + 545. iT - 1.61e5T^{2} \)
13 \( 1 + 885. iT - 3.71e5T^{2} \)
17 \( 1 - 2.30e3T + 1.41e6T^{2} \)
23 \( 1 + 3.45e3iT - 6.43e6T^{2} \)
29 \( 1 - 2.57e3iT - 2.05e7T^{2} \)
31 \( 1 - 4.68e3T + 2.86e7T^{2} \)
37 \( 1 - 9.75e3iT - 6.93e7T^{2} \)
41 \( 1 - 8.78e3iT - 1.15e8T^{2} \)
43 \( 1 + 1.64e4iT - 1.47e8T^{2} \)
47 \( 1 - 7.32e3iT - 2.29e8T^{2} \)
53 \( 1 - 1.13e4iT - 4.18e8T^{2} \)
59 \( 1 - 2.59e4T + 7.14e8T^{2} \)
61 \( 1 - 4.29e4T + 8.44e8T^{2} \)
67 \( 1 - 7.12e3T + 1.35e9T^{2} \)
71 \( 1 + 1.83e4T + 1.80e9T^{2} \)
73 \( 1 + 1.36e4T + 2.07e9T^{2} \)
79 \( 1 - 4.31e4T + 3.07e9T^{2} \)
83 \( 1 - 2.69e3iT - 3.93e9T^{2} \)
89 \( 1 - 4.20e4iT - 5.58e9T^{2} \)
97 \( 1 - 3.99e4iT - 8.58e9T^{2} \)
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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

−13.23504315945399819007557828968, −12.10538944287083984723129172122, −10.64561929292519238195795573427, −9.931783250854467767939960211757, −8.450323666635652371536298793587, −8.032227647542122300256252861477, −5.63854078684938598083564720848, −3.64192377880346063605316033356, −2.74360410413167778772733532427, −0.841747146174499306978081140577, 1.96199617257402929504846356523, 4.03259128451700902793248274203, 5.62147300265821034832148511422, 7.11243473450099372982375347695, 8.118808419845769693057576698839, 9.324501542589849209523024454258, 9.839730515435861045270915971720, 11.92620227169354187732655205814, 13.28099873345783987584029140920, 14.25264332184360061825902458270

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