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Properties

Label 2-76-76.75-c3-0-22
Degree $2$
Conductor $76$
Sign $0.429 + 0.902i$
Analytic cond. $4.48414$
Root an. cond. $2.11758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.60 − 2.33i)2-s + 3.54·3-s + (−2.87 − 7.46i)4-s + 18.5·5-s + (5.67 − 8.26i)6-s + 16.5i·7-s + (−22.0 − 5.26i)8-s − 14.4·9-s + (29.7 − 43.2i)10-s + 9.00i·11-s + (−10.1 − 26.4i)12-s − 60.5i·13-s + (38.6 + 26.5i)14-s + 65.8·15-s + (−47.5 + 42.8i)16-s + 1.33·17-s + ⋯
L(s)  = 1  + (0.566 − 0.824i)2-s + 0.682·3-s + (−0.358 − 0.933i)4-s + 1.66·5-s + (0.386 − 0.562i)6-s + 0.894i·7-s + (−0.972 − 0.232i)8-s − 0.534·9-s + (0.940 − 1.36i)10-s + 0.246i·11-s + (−0.244 − 0.636i)12-s − 1.29i·13-s + (0.737 + 0.506i)14-s + 1.13·15-s + (−0.742 + 0.669i)16-s + 0.0189·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.429 + 0.902i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.429 + 0.902i$
Analytic conductor: \(4.48414\)
Root analytic conductor: \(2.11758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :3/2),\ 0.429 + 0.902i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.21739 - 1.40003i\)
\(L(\frac12)\) \(\approx\) \(2.21739 - 1.40003i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.60 + 2.33i)T \)
19 \( 1 + (82.5 - 6.40i)T \)
good3 \( 1 - 3.54T + 27T^{2} \)
5 \( 1 - 18.5T + 125T^{2} \)
7 \( 1 - 16.5iT - 343T^{2} \)
11 \( 1 - 9.00iT - 1.33e3T^{2} \)
13 \( 1 + 60.5iT - 2.19e3T^{2} \)
17 \( 1 - 1.33T + 4.91e3T^{2} \)
23 \( 1 - 182. iT - 1.21e4T^{2} \)
29 \( 1 + 61.1iT - 2.43e4T^{2} \)
31 \( 1 - 111.T + 2.97e4T^{2} \)
37 \( 1 - 160. iT - 5.06e4T^{2} \)
41 \( 1 - 386. iT - 6.89e4T^{2} \)
43 \( 1 + 300. iT - 7.95e4T^{2} \)
47 \( 1 + 206. iT - 1.03e5T^{2} \)
53 \( 1 + 544. iT - 1.48e5T^{2} \)
59 \( 1 - 444.T + 2.05e5T^{2} \)
61 \( 1 + 252.T + 2.26e5T^{2} \)
67 \( 1 + 530.T + 3.00e5T^{2} \)
71 \( 1 + 354.T + 3.57e5T^{2} \)
73 \( 1 - 343.T + 3.89e5T^{2} \)
79 \( 1 + 26.6T + 4.93e5T^{2} \)
83 \( 1 + 1.38e3iT - 5.71e5T^{2} \)
89 \( 1 + 977. iT - 7.04e5T^{2} \)
97 \( 1 - 1.18e3iT - 9.12e5T^{2} \)
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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

−13.54494495142914887105016388094, −13.07772112000766899997678429149, −11.75461321107086451840458960171, −10.31885689093019925675701524605, −9.518453054063246786705662206413, −8.510013068900436008126120856405, −6.07020664608307736567654453513, −5.31599791895138165795736670210, −3.02564173464712585634470014818, −1.98143548563893479346338419638, 2.47745941520211610022386705158, 4.34175092704471211292013348417, 5.94781578781011329012788288230, 6.88750583088137458518876548390, 8.525987520683244951793571377857, 9.343161168159762780510761673930, 10.76770560714850044570218113321, 12.55496518543859869014804843153, 13.73945469466833595477086347913, 14.00026034896654033186826232945

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