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Properties

Label 2-71-71.25-c1-0-1
Degree $2$
Conductor $71$
Sign $0.0171 - 0.999i$
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  + (0.363 − 0.264i)2-s + (−2.65 + 1.93i)3-s + (−0.555 + 1.70i)4-s + (0.225 + 0.694i)5-s + (−0.456 + 1.40i)6-s + (1.83 − 1.33i)7-s + (0.527 + 1.62i)8-s + (2.40 − 7.41i)9-s + (0.265 + 0.192i)10-s + (−2.07 + 1.51i)11-s + (−1.82 − 5.61i)12-s + (3.29 + 2.39i)13-s + (0.314 − 0.969i)14-s + (−1.94 − 1.41i)15-s + (−2.28 − 1.66i)16-s + (3.62 + 2.63i)17-s + ⋯
L(s)  = 1  + (0.257 − 0.186i)2-s + (−1.53 + 1.11i)3-s + (−0.277 + 0.854i)4-s + (0.100 + 0.310i)5-s + (−0.186 + 0.573i)6-s + (0.693 − 0.503i)7-s + (0.186 + 0.573i)8-s + (0.802 − 2.47i)9-s + (0.0839 + 0.0610i)10-s + (−0.626 + 0.455i)11-s + (−0.526 − 1.62i)12-s + (0.914 + 0.664i)13-s + (0.0841 − 0.259i)14-s + (−0.501 − 0.364i)15-s + (−0.572 − 0.415i)16-s + (0.878 + 0.638i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0171 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0171 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $0.0171 - 0.999i$
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: \((2,\ 71,\ (\ :1/2),\ 0.0171 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.488057 + 0.479753i\)
\(L(\frac12)\) \(\approx\) \(0.488057 + 0.479753i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (-5.05 + 6.73i)T \)
good2 \( 1 + (-0.363 + 0.264i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (2.65 - 1.93i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.225 - 0.694i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.83 + 1.33i)T + (2.16 - 6.65i)T^{2} \)
11 \( 1 + (2.07 - 1.51i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-3.29 - 2.39i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.62 - 2.63i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-0.776 + 0.564i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.09T + 23T^{2} \)
29 \( 1 + (2.99 + 9.21i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-4.51 + 3.28i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + 1.47T + 37T^{2} \)
41 \( 1 + 5.12T + 41T^{2} \)
43 \( 1 + (-2.48 - 7.64i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-1.00 - 0.731i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.352 + 1.08i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.636 + 1.95i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-0.513 - 0.373i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.05 + 12.4i)T + (-54.2 - 39.3i)T^{2} \)
73 \( 1 + (-5.60 + 4.07i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (1.57 + 4.83i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.182 + 0.561i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-2.46 - 7.59i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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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

−15.14562561533971763447741721442, −13.80943110082683352018570954017, −12.42848169837746820006150209390, −11.52211472875972213435899513672, −10.75976114532081275226345906740, −9.686958335702398184836764030669, −7.949640138395677529064110174889, −6.25736231137653185668429714456, −4.81422374269933414350600938583, −3.89445103871393254767801166115, 1.23039057956880142318659054125, 5.32064139075623156978222708247, 5.48591542683708843877672380921, 6.94277950869212870053915691963, 8.386038037409511808759854617750, 10.35766814256969912890817546003, 11.17199178122481874255906581786, 12.29473171371998910499869170696, 13.22010525639183112968086088331, 14.14053294454848586288926352441

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