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Properties

Label 2-76-76.75-c5-0-5
Degree $2$
Conductor $76$
Sign $-0.743 - 0.669i$
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  + (−2.99 + 4.79i)2-s − 24.6·3-s + (−14.0 − 28.7i)4-s + 37.3·5-s + (73.8 − 118. i)6-s − 35.5i·7-s + (180. + 19.1i)8-s + 362.·9-s + (−112. + 179. i)10-s − 184. i·11-s + (344. + 708. i)12-s − 119. i·13-s + (170. + 106. i)14-s − 919.·15-s + (−631. + 806. i)16-s + 152.·17-s + ⋯
L(s)  = 1  + (−0.530 + 0.847i)2-s − 1.57·3-s + (−0.437 − 0.899i)4-s + 0.668·5-s + (0.837 − 1.33i)6-s − 0.274i·7-s + (0.994 + 0.105i)8-s + 1.49·9-s + (−0.354 + 0.566i)10-s − 0.460i·11-s + (0.691 + 1.41i)12-s − 0.195i·13-s + (0.232 + 0.145i)14-s − 1.05·15-s + (−0.616 + 0.787i)16-s + 0.128·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.743 - 0.669i$
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: \((2,\ 76,\ (\ :5/2),\ -0.743 - 0.669i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.168050 + 0.437710i\)
\(L(\frac12)\) \(\approx\) \(0.168050 + 0.437710i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.99 - 4.79i)T \)
19 \( 1 + (1.45e3 - 590. i)T \)
good3 \( 1 + 24.6T + 243T^{2} \)
5 \( 1 - 37.3T + 3.12e3T^{2} \)
7 \( 1 + 35.5iT - 1.68e4T^{2} \)
11 \( 1 + 184. iT - 1.61e5T^{2} \)
13 \( 1 + 119. iT - 3.71e5T^{2} \)
17 \( 1 - 152.T + 1.41e6T^{2} \)
23 \( 1 - 1.58e3iT - 6.43e6T^{2} \)
29 \( 1 + 54.3iT - 2.05e7T^{2} \)
31 \( 1 - 4.26e3T + 2.86e7T^{2} \)
37 \( 1 - 1.21e4iT - 6.93e7T^{2} \)
41 \( 1 - 5.66e3iT - 1.15e8T^{2} \)
43 \( 1 - 1.92e4iT - 1.47e8T^{2} \)
47 \( 1 - 2.66e4iT - 2.29e8T^{2} \)
53 \( 1 + 2.94e4iT - 4.18e8T^{2} \)
59 \( 1 + 2.28e4T + 7.14e8T^{2} \)
61 \( 1 + 3.17e4T + 8.44e8T^{2} \)
67 \( 1 - 2.17e4T + 1.35e9T^{2} \)
71 \( 1 + 1.28e4T + 1.80e9T^{2} \)
73 \( 1 - 3.33e4T + 2.07e9T^{2} \)
79 \( 1 + 5.70e4T + 3.07e9T^{2} \)
83 \( 1 - 1.47e4iT - 3.93e9T^{2} \)
89 \( 1 - 1.38e5iT - 5.58e9T^{2} \)
97 \( 1 + 3.42e4iT - 8.58e9T^{2} \)
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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

−13.98623370192482931921585078300, −12.93102112390559886107893842409, −11.47037711264902795825380615071, −10.50211720589684161743627194439, −9.645686460466095512234454916585, −8.001016634240254907400069576516, −6.49353728064982607845095072489, −5.88556207113524121264177591359, −4.69101997867082112485633943123, −1.18188594245311542593790655115, 0.34429413873805626347043993941, 2.04638387475495381874123405444, 4.40131441651408522699747531671, 5.74768878298824109753116575668, 7.08008412383838687182029701397, 8.878340495916674021727537536707, 10.13948946641877184257453535701, 10.81540124075977542660793271622, 11.95816524203961127392226182351, 12.57542153452946408498701876957

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