Properties

Label 2-76-76.71-c1-0-0
Degree $2$
Conductor $76$
Sign $-0.481 - 0.876i$
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  + (−0.0238 + 1.41i)2-s + (−0.306 + 1.73i)3-s + (−1.99 − 0.0675i)4-s + (−0.220 − 0.184i)5-s + (−2.45 − 0.475i)6-s + (0.588 − 0.339i)7-s + (0.143 − 2.82i)8-s + (−0.110 − 0.0403i)9-s + (0.266 − 0.306i)10-s + (3.85 + 2.22i)11-s + (0.730 − 3.45i)12-s + (−3.41 + 0.602i)13-s + (0.466 + 0.840i)14-s + (0.388 − 0.326i)15-s + (3.99 + 0.270i)16-s + (4.15 − 1.51i)17-s + ⋯
L(s)  = 1  + (−0.0168 + 0.999i)2-s + (−0.177 + 1.00i)3-s + (−0.999 − 0.0337i)4-s + (−0.0984 − 0.0826i)5-s + (−1.00 − 0.193i)6-s + (0.222 − 0.128i)7-s + (0.0506 − 0.998i)8-s + (−0.0369 − 0.0134i)9-s + (0.0842 − 0.0970i)10-s + (1.16 + 0.670i)11-s + (0.210 − 0.997i)12-s + (−0.947 + 0.166i)13-s + (0.124 + 0.224i)14-s + (0.100 − 0.0842i)15-s + (0.997 + 0.0675i)16-s + (1.00 − 0.366i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.481 - 0.876i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $-0.481 - 0.876i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ -0.481 - 0.876i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.428108 + 0.723909i\)
\(L(\frac12)\) \(\approx\) \(0.428108 + 0.723909i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0238 - 1.41i)T \)
19 \( 1 + (1.76 + 3.98i)T \)
good3 \( 1 + (0.306 - 1.73i)T + (-2.81 - 1.02i)T^{2} \)
5 \( 1 + (0.220 + 0.184i)T + (0.868 + 4.92i)T^{2} \)
7 \( 1 + (-0.588 + 0.339i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3.85 - 2.22i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.41 - 0.602i)T + (12.2 - 4.44i)T^{2} \)
17 \( 1 + (-4.15 + 1.51i)T + (13.0 - 10.9i)T^{2} \)
23 \( 1 + (0.347 + 0.413i)T + (-3.99 + 22.6i)T^{2} \)
29 \( 1 + (-1.03 + 2.85i)T + (-22.2 - 18.6i)T^{2} \)
31 \( 1 + (5.24 + 9.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 8.82iT - 37T^{2} \)
41 \( 1 + (-1.85 - 0.326i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (3.49 - 4.16i)T + (-7.46 - 42.3i)T^{2} \)
47 \( 1 + (0.419 - 1.15i)T + (-36.0 - 30.2i)T^{2} \)
53 \( 1 + (6.41 + 7.64i)T + (-9.20 + 52.1i)T^{2} \)
59 \( 1 + (-4.20 + 1.53i)T + (45.1 - 37.9i)T^{2} \)
61 \( 1 + (-6.04 + 5.07i)T + (10.5 - 60.0i)T^{2} \)
67 \( 1 + (-4.66 - 1.69i)T + (51.3 + 43.0i)T^{2} \)
71 \( 1 + (-6.81 - 5.72i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + (-0.591 + 3.35i)T + (-68.5 - 24.9i)T^{2} \)
79 \( 1 + (-1.43 + 8.15i)T + (-74.2 - 27.0i)T^{2} \)
83 \( 1 + (8.64 - 4.99i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (10.6 - 1.87i)T + (83.6 - 30.4i)T^{2} \)
97 \( 1 + (1.24 + 3.40i)T + (-74.3 + 62.3i)T^{2} \)
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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

−14.91810791010423193636328732645, −14.33727974442257449906474814013, −12.87152380088899514267567562119, −11.58488113492748735785934248400, −9.920090082700821629189304421466, −9.449149842906507641370329630240, −7.88331497042823728506483223749, −6.63749240959283523245845164483, −4.98487103009129182876384471820, −4.12826679791545416512166739710, 1.56380952101430898256648814964, 3.65054189296029366959591427245, 5.58737017092271667078729815019, 7.22947395669401901726018188686, 8.520376965713788264721548021130, 9.813806174044964910925976625963, 11.10108103677465377820961992190, 12.23158396072873783966811619881, 12.60291725289286122577605466559, 14.02119626517136675207708228943

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