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Properties

Label 2-76-76.31-c1-0-5
Degree $2$
Conductor $76$
Sign $0.448 + 0.893i$
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  + (−1.05 + 0.947i)2-s + (−0.982 − 1.70i)3-s + (0.205 − 1.98i)4-s + (−0.349 − 0.605i)5-s + (2.64 + 0.855i)6-s − 3.80i·7-s + (1.66 + 2.28i)8-s + (−0.430 + 0.744i)9-s + (0.940 + 0.304i)10-s + 2.16i·11-s + (−3.58 + 1.60i)12-s + (1.16 + 0.672i)13-s + (3.60 + 3.99i)14-s + (−0.686 + 1.18i)15-s + (−3.91 − 0.816i)16-s + (−1.89 − 3.28i)17-s + ⋯
L(s)  = 1  + (−0.742 + 0.669i)2-s + (−0.567 − 0.982i)3-s + (0.102 − 0.994i)4-s + (−0.156 − 0.270i)5-s + (1.07 + 0.349i)6-s − 1.43i·7-s + (0.590 + 0.807i)8-s + (−0.143 + 0.248i)9-s + (0.297 + 0.0963i)10-s + 0.653i·11-s + (−1.03 + 0.463i)12-s + (0.323 + 0.186i)13-s + (0.962 + 1.06i)14-s + (−0.177 + 0.307i)15-s + (−0.978 − 0.204i)16-s + (−0.459 − 0.796i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.448 + 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.448 + 0.893i$
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: \((2,\ 76,\ (\ :1/2),\ 0.448 + 0.893i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.467267 - 0.288244i\)
\(L(\frac12)\) \(\approx\) \(0.467267 - 0.288244i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.05 - 0.947i)T \)
19 \( 1 + (1.62 - 4.04i)T \)
good3 \( 1 + (0.982 + 1.70i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.349 + 0.605i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 3.80iT - 7T^{2} \)
11 \( 1 - 2.16iT - 11T^{2} \)
13 \( 1 + (-1.16 - 0.672i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.89 + 3.28i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-4.89 - 2.82i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-8.65 - 4.99i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.76T + 31T^{2} \)
37 \( 1 + 1.31iT - 37T^{2} \)
41 \( 1 + (7.58 - 4.37i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.35 + 3.08i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.06 + 1.18i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.41 + 3.12i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.28 + 5.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.951 - 1.64i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.60 + 4.51i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-4.86 - 8.41i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-3.38 - 5.86i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 1.55iT - 83T^{2} \)
89 \( 1 + (1.43 + 0.827i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (9.10 - 5.25i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.25699115441116290651749577973, −13.38210119938215645987422134660, −12.12797126675416048576204957566, −10.88642986955314393317391642646, −9.842621785452777032766107084687, −8.284388670214223892161857904637, −7.12554531923846235488567751283, −6.54570348303357327923054048755, −4.70844331613371437055541721131, −1.10084741136985766355728384195, 2.87644911167437973622970824810, 4.68337436167325430767561222701, 6.34315136566543115064473975028, 8.350538943300245573449152767511, 9.162013778491598109209612138371, 10.46963876411199851824296415653, 11.15964474669378050896356573944, 12.10798096137750464126329442054, 13.33393222986930226870391886574, 15.30347449769642657525911282289

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