Properties

Label 2-76-76.3-c1-0-6
Degree $2$
Conductor $76$
Sign $0.943 + 0.330i$
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  + (1.33 − 0.462i)2-s + (−0.855 + 0.311i)3-s + (1.57 − 1.23i)4-s + (−0.00805 + 0.0456i)5-s + (−0.999 + 0.811i)6-s + (−1.20 + 0.698i)7-s + (1.52 − 2.37i)8-s + (−1.66 + 1.39i)9-s + (0.0103 + 0.0647i)10-s + (−1.13 − 0.655i)11-s + (−0.960 + 1.54i)12-s + (−0.681 + 1.87i)13-s + (−1.29 + 1.49i)14-s + (−0.00733 − 0.0416i)15-s + (0.944 − 3.88i)16-s + (−0.910 − 0.763i)17-s + ⋯
L(s)  = 1  + (0.945 − 0.326i)2-s + (−0.494 + 0.179i)3-s + (0.786 − 0.618i)4-s + (−0.00360 + 0.0204i)5-s + (−0.408 + 0.331i)6-s + (−0.457 + 0.263i)7-s + (0.540 − 0.841i)8-s + (−0.554 + 0.465i)9-s + (0.00327 + 0.0204i)10-s + (−0.342 − 0.197i)11-s + (−0.277 + 0.446i)12-s + (−0.189 + 0.519i)13-s + (−0.345 + 0.398i)14-s + (−0.00189 − 0.0107i)15-s + (0.236 − 0.971i)16-s + (−0.220 − 0.185i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.943 + 0.330i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.943 + 0.330i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.24489 - 0.211622i\)
\(L(\frac12)\) \(\approx\) \(1.24489 - 0.211622i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.33 + 0.462i)T \)
19 \( 1 + (4.35 - 0.189i)T \)
good3 \( 1 + (0.855 - 0.311i)T + (2.29 - 1.92i)T^{2} \)
5 \( 1 + (0.00805 - 0.0456i)T + (-4.69 - 1.71i)T^{2} \)
7 \( 1 + (1.20 - 0.698i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.13 + 0.655i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.681 - 1.87i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (0.910 + 0.763i)T + (2.95 + 16.7i)T^{2} \)
23 \( 1 + (-8.42 + 1.48i)T + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (3.45 + 4.11i)T + (-5.03 + 28.5i)T^{2} \)
31 \( 1 + (-3.06 - 5.31i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 4.41iT - 37T^{2} \)
41 \( 1 + (2.39 + 6.59i)T + (-31.4 + 26.3i)T^{2} \)
43 \( 1 + (0.406 + 0.0716i)T + (40.4 + 14.7i)T^{2} \)
47 \( 1 + (-5.36 - 6.39i)T + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-5.17 + 0.913i)T + (49.8 - 18.1i)T^{2} \)
59 \( 1 + (8.98 + 7.53i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-2.12 - 12.0i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (8.40 - 7.05i)T + (11.6 - 65.9i)T^{2} \)
71 \( 1 + (-1.06 + 6.02i)T + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (6.09 - 2.21i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (10.3 - 3.76i)T + (60.5 - 50.7i)T^{2} \)
83 \( 1 + (0.635 - 0.367i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (4.73 - 12.9i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-5.45 + 6.50i)T + (-16.8 - 95.5i)T^{2} \)
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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

−14.37387350794242419908863677803, −13.27542004043590307162974239675, −12.36758753011636458644042084584, −11.20158984245098462252501228296, −10.51750178520895940886724148835, −8.936802971783965012494190749979, −7.01069054462491217853735546170, −5.78881287767559814098908073543, −4.62925966715804495688326973512, −2.75678634181994047922049745320, 3.10179701046400927806648217683, 4.87122619486253653997501314417, 6.16025150411814929290033501919, 7.15496498098796896961071042356, 8.689130703763575923699880780977, 10.51568530863719401194374868783, 11.51064852931756577470484119410, 12.68688230112079678627047508636, 13.25601674650874914137205449084, 14.73571911573782770811051161993

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