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Properties

Label 2-76-76.31-c1-0-6
Degree $2$
Conductor $76$
Sign $0.902 + 0.430i$
Analytic cond. $0.606863$
Root an. cond. $0.779014$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.35 + 0.403i)2-s + (−1.42 − 2.47i)3-s + (1.67 + 1.09i)4-s + (−0.139 − 0.242i)5-s + (−0.935 − 3.92i)6-s + 1.55i·7-s + (1.82 + 2.15i)8-s + (−2.57 + 4.45i)9-s + (−0.0917 − 0.385i)10-s + 2.44i·11-s + (0.317 − 5.69i)12-s + (−5.47 − 3.16i)13-s + (−0.627 + 2.10i)14-s + (−0.399 + 0.691i)15-s + (1.60 + 3.66i)16-s + (2.18 + 3.78i)17-s + ⋯
L(s)  = 1  + (0.958 + 0.285i)2-s + (−0.823 − 1.42i)3-s + (0.836 + 0.547i)4-s + (−0.0625 − 0.108i)5-s + (−0.382 − 1.60i)6-s + 0.586i·7-s + (0.645 + 0.763i)8-s + (−0.857 + 1.48i)9-s + (−0.0290 − 0.121i)10-s + 0.735i·11-s + (0.0916 − 1.64i)12-s + (−1.51 − 0.876i)13-s + (−0.167 + 0.562i)14-s + (−0.103 + 0.178i)15-s + (0.400 + 0.916i)16-s + (0.529 + 0.917i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.902 + 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.902 + 0.430i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.902 + 0.430i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18271 - 0.267332i\)
\(L(\frac12)\) \(\approx\) \(1.18271 - 0.267332i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.35 - 0.403i)T \)
19 \( 1 + (-3.17 + 2.99i)T \)
good3 \( 1 + (1.42 + 2.47i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.139 + 0.242i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 1.55iT - 7T^{2} \)
11 \( 1 - 2.44iT - 11T^{2} \)
13 \( 1 + (5.47 + 3.16i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.18 - 3.78i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (5.71 + 3.30i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.695 - 0.401i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.42T + 31T^{2} \)
37 \( 1 + 4.74iT - 37T^{2} \)
41 \( 1 + (5.84 - 3.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.146 - 0.0845i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.19 + 0.691i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.25 - 2.45i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.05 + 1.81i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.58 - 2.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.00 + 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1.74 - 3.02i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.43 + 9.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.00 + 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.99iT - 83T^{2} \)
89 \( 1 + (-7.76 - 4.48i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.77 - 1.02i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.31688501965218580791360188964, −13.05219341102750161553230017546, −12.24756423969305952928374146001, −11.98736732265771807874944005105, −10.37002960980105190475046673100, −8.056225720746456316155797589670, −7.17199921744983239652552839350, −6.03157357361439118424871611415, −4.96623737727796089582688977149, −2.36775801665520762926267996118, 3.46696304127139788796125941931, 4.69477559169631948134796612158, 5.65959240621497333581773984732, 7.21843858331729021868962132983, 9.654242134399059339076241709172, 10.26091541110453733223911721316, 11.51024252707018815353350553237, 11.97974217022629416659456963391, 13.78451726080033103566549165998, 14.53957228589133853088491113428

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