Properties

Label 2-76-76.51-c1-0-2
Degree $2$
Conductor $76$
Sign $0.762 - 0.646i$
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  + (−1.19 + 0.750i)2-s + (2.80 + 1.02i)3-s + (0.874 − 1.79i)4-s + (−0.165 − 0.936i)5-s + (−4.12 + 0.880i)6-s + (−2.67 − 1.54i)7-s + (0.301 + 2.81i)8-s + (4.52 + 3.79i)9-s + (0.900 + 0.998i)10-s + (−3.39 + 1.95i)11-s + (4.28 − 4.15i)12-s + (−0.284 − 0.781i)13-s + (4.36 − 0.155i)14-s + (0.492 − 2.79i)15-s + (−2.47 − 3.14i)16-s + (−3.14 + 2.63i)17-s + ⋯
L(s)  = 1  + (−0.847 + 0.530i)2-s + (1.61 + 0.589i)3-s + (0.437 − 0.899i)4-s + (−0.0738 − 0.418i)5-s + (−1.68 + 0.359i)6-s + (−1.01 − 0.583i)7-s + (0.106 + 0.994i)8-s + (1.50 + 1.26i)9-s + (0.284 + 0.315i)10-s + (−1.02 + 0.590i)11-s + (1.23 − 1.19i)12-s + (−0.0789 − 0.216i)13-s + (1.16 − 0.0415i)14-s + (0.127 − 0.721i)15-s + (−0.617 − 0.786i)16-s + (−0.762 + 0.639i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.762 - 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(76\)    =    \(2^{2} \cdot 19\)
Sign: $0.762 - 0.646i$
Analytic conductor: \(0.606863\)
Root analytic conductor: \(0.779014\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{76} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 76,\ (\ :1/2),\ 0.762 - 0.646i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862922 + 0.316497i\)
\(L(\frac12)\) \(\approx\) \(0.862922 + 0.316497i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.19 - 0.750i)T \)
19 \( 1 + (0.473 + 4.33i)T \)
good3 \( 1 + (-2.80 - 1.02i)T + (2.29 + 1.92i)T^{2} \)
5 \( 1 + (0.165 + 0.936i)T + (-4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.67 + 1.54i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (3.39 - 1.95i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.284 + 0.781i)T + (-9.95 + 8.35i)T^{2} \)
17 \( 1 + (3.14 - 2.63i)T + (2.95 - 16.7i)T^{2} \)
23 \( 1 + (-2.45 - 0.432i)T + (21.6 + 7.86i)T^{2} \)
29 \( 1 + (-1.95 + 2.32i)T + (-5.03 - 28.5i)T^{2} \)
31 \( 1 + (-0.560 + 0.970i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 11.0iT - 37T^{2} \)
41 \( 1 + (-1.74 + 4.80i)T + (-31.4 - 26.3i)T^{2} \)
43 \( 1 + (-2.92 + 0.515i)T + (40.4 - 14.7i)T^{2} \)
47 \( 1 + (2.23 - 2.66i)T + (-8.16 - 46.2i)T^{2} \)
53 \( 1 + (4.27 + 0.753i)T + (49.8 + 18.1i)T^{2} \)
59 \( 1 + (0.677 - 0.568i)T + (10.2 - 58.1i)T^{2} \)
61 \( 1 + (1.74 - 9.90i)T + (-57.3 - 20.8i)T^{2} \)
67 \( 1 + (-6.32 - 5.30i)T + (11.6 + 65.9i)T^{2} \)
71 \( 1 + (1.66 + 9.42i)T + (-66.7 + 24.2i)T^{2} \)
73 \( 1 + (-2.43 - 0.887i)T + (55.9 + 46.9i)T^{2} \)
79 \( 1 + (3.07 + 1.12i)T + (60.5 + 50.7i)T^{2} \)
83 \( 1 + (-10.7 - 6.18i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (0.254 + 0.698i)T + (-68.1 + 57.2i)T^{2} \)
97 \( 1 + (6.10 + 7.27i)T + (-16.8 + 95.5i)T^{2} \)
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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

−15.08153813913565297844984280359, −13.69987055874065279287173388792, −12.95259656953737973726111171352, −10.63635513224767208164607912713, −9.852301269297246121149570650340, −8.940266499832442026891466295986, −8.015313112818155367119303398247, −6.85843235702874012926412890047, −4.64552208143904293682755852171, −2.71884480184188186654903510753, 2.47790865015715896960881301388, 3.33504474049100923509935641314, 6.67998103575862661122546036143, 7.76971576633982674271008788913, 8.799554126332268348946096443021, 9.568506145485025754672736512217, 10.86546748797892587997714221345, 12.51844615899881467970301399155, 13.10343127541715042588841298805, 14.28040509593973077779637104612

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