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Properties

Label 2-76-76.31-c1-0-2
Degree $2$
Conductor $76$
Sign $0.997 - 0.0654i$
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  + (−1.16 − 0.801i)2-s + (0.305 + 0.528i)3-s + (0.713 + 1.86i)4-s + (1.59 + 2.75i)5-s + (0.0683 − 0.860i)6-s − 2.36i·7-s + (0.666 − 2.74i)8-s + (1.31 − 2.27i)9-s + (0.357 − 4.49i)10-s + 5.46i·11-s + (−0.769 + 0.947i)12-s + (−2.31 − 1.33i)13-s + (−1.89 + 2.75i)14-s + (−0.971 + 1.68i)15-s + (−2.98 + 2.66i)16-s + (−0.552 − 0.957i)17-s + ⋯
L(s)  = 1  + (−0.823 − 0.567i)2-s + (0.176 + 0.305i)3-s + (0.356 + 0.934i)4-s + (0.712 + 1.23i)5-s + (0.0279 − 0.351i)6-s − 0.893i·7-s + (0.235 − 0.971i)8-s + (0.437 − 0.758i)9-s + (0.112 − 1.42i)10-s + 1.64i·11-s + (−0.222 + 0.273i)12-s + (−0.643 − 0.371i)13-s + (−0.506 + 0.735i)14-s + (−0.250 + 0.434i)15-s + (−0.745 + 0.666i)16-s + (−0.134 − 0.232i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.997 - 0.0654i$
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: \((2,\ 76,\ (\ :1/2),\ 0.997 - 0.0654i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.755813 + 0.0247479i\)
\(L(\frac12)\) \(\approx\) \(0.755813 + 0.0247479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.16 + 0.801i)T \)
19 \( 1 + (1.37 + 4.13i)T \)
good3 \( 1 + (-0.305 - 0.528i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.59 - 2.75i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + 2.36iT - 7T^{2} \)
11 \( 1 - 5.46iT - 11T^{2} \)
13 \( 1 + (2.31 + 1.33i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.552 + 0.957i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (2.46 + 1.42i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.63 + 3.25i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.01T + 31T^{2} \)
37 \( 1 + 0.450iT - 37T^{2} \)
41 \( 1 + (-0.336 + 0.194i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.96 + 2.86i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.91 + 1.68i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.53 + 2.03i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.82 - 11.8i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.77 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4.27 + 7.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (1.07 + 1.86i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-3.91 - 6.78i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.57 - 9.65i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 4.14iT - 83T^{2} \)
89 \( 1 + (-4.19 - 2.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (0.641 - 0.370i)T + (48.5 - 84.0i)T^{2} \)
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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

−14.70015038468017133355047566024, −13.33105398491909653991869403362, −12.21068856371147162977902739186, −10.78752174255305621964488365132, −10.03927368218912072177487942972, −9.415568868636859779214341719110, −7.39833747936368202127942397879, −6.79563006028509201607431902235, −4.11449578699098636743939742081, −2.42457295970212046392523880919, 1.84003715972310922104809778590, 5.19327343994532858622576420564, 6.09022626546766529175484692074, 7.915686060203221249955806280500, 8.740600389324995148066977651170, 9.622102587313812964211423052639, 11.01772024881656590852205746067, 12.45011887221669007418638342909, 13.54705535539065417067436708988, 14.50857859013455925743586697868

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