Properties

Label 2-71-71.25-c1-0-3
Degree $2$
Conductor $71$
Sign $0.980 + 0.197i$
Analytic cond. $0.566937$
Root an. cond. $0.752952$
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  + (−2.18 + 1.58i)2-s + (1.68 − 1.22i)3-s + (1.63 − 5.04i)4-s + (−0.994 − 3.06i)5-s + (−1.73 + 5.34i)6-s + (0.649 − 0.472i)7-s + (2.75 + 8.49i)8-s + (0.410 − 1.26i)9-s + (7.03 + 5.11i)10-s + (−1.76 + 1.27i)11-s + (−3.41 − 10.4i)12-s + (2.15 + 1.56i)13-s + (−0.670 + 2.06i)14-s + (−5.41 − 3.93i)15-s + (−10.9 − 7.94i)16-s + (1.63 + 1.18i)17-s + ⋯
L(s)  = 1  + (−1.54 + 1.12i)2-s + (0.971 − 0.706i)3-s + (0.819 − 2.52i)4-s + (−0.444 − 1.36i)5-s + (−0.709 + 2.18i)6-s + (0.245 − 0.178i)7-s + (0.975 + 3.00i)8-s + (0.136 − 0.421i)9-s + (2.22 + 1.61i)10-s + (−0.531 + 0.385i)11-s + (−0.984 − 3.02i)12-s + (0.598 + 0.435i)13-s + (−0.179 + 0.551i)14-s + (−1.39 − 1.01i)15-s + (−2.73 − 1.98i)16-s + (0.396 + 0.288i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.980 + 0.197i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $0.980 + 0.197i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ 0.980 + 0.197i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.585838 - 0.0583859i\)
\(L(\frac12)\) \(\approx\) \(0.585838 - 0.0583859i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (-5.71 - 6.19i)T \)
good2 \( 1 + (2.18 - 1.58i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-1.68 + 1.22i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (0.994 + 3.06i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-0.649 + 0.472i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (1.76 - 1.27i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-2.15 - 1.56i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.63 - 1.18i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.21 - 0.881i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.57T + 23T^{2} \)
29 \( 1 + (0.0382 + 0.117i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.464 - 0.337i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + 5.83T + 37T^{2} \)
41 \( 1 + 8.59T + 41T^{2} \)
43 \( 1 + (-0.557 - 1.71i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-8.39 - 6.09i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (4.03 + 12.4i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.69 + 11.3i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.35 - 0.981i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.173 - 0.535i)T + (-54.2 - 39.3i)T^{2} \)
73 \( 1 + (1.21 - 0.884i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (2.21 + 6.81i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.61 - 11.1i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.52 - 4.69i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + 12.6T + 97T^{2} \)
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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

−14.96665926704508049801505282148, −13.86512326820464480449117596875, −12.65901672571083566657251741708, −10.93637502891101082222291953681, −9.432687275123693049638160376291, −8.510736073779759665929404219902, −8.033932831173493298030159566756, −6.93510093975587795625800909165, −5.14035805562653899261156169725, −1.47718653552201638364070714439, 2.77049752772748932450342951887, 3.51717706177472345292274758075, 7.13256483514187258440730088721, 8.249459292170318397361368348086, 9.091736003449581012332661479482, 10.37846253393884593774311243386, 10.83520852540032991549175309401, 11.96683997160361439639792443114, 13.57094203074318149436250068236, 15.03913699544302363585771580008

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