↑

Properties

Label 2-71-71.5-c1-0-1
Degree $2$
Conductor $71$
Sign $0.949 + 0.312i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.561 − 1.72i)2-s + (0.906 + 2.79i)3-s + (−1.05 + 0.763i)4-s + (1.21 + 0.879i)5-s + (4.31 − 3.13i)6-s + (−0.388 − 1.19i)7-s + (−1.03 − 0.749i)8-s + (−4.54 + 3.30i)9-s + (0.839 − 2.58i)10-s + (−1.22 − 3.77i)11-s + (−3.08 − 2.23i)12-s + (−0.173 + 0.535i)13-s + (−1.84 + 1.34i)14-s + (−1.35 + 4.17i)15-s + (−1.51 + 4.67i)16-s + (1.87 − 5.77i)17-s + ⋯
L(s)  = 1  + (−0.396 − 1.22i)2-s + (0.523 + 1.61i)3-s + (−0.525 + 0.381i)4-s + (0.541 + 0.393i)5-s + (1.76 − 1.27i)6-s + (−0.146 − 0.451i)7-s + (−0.364 − 0.264i)8-s + (−1.51 + 1.10i)9-s + (0.265 − 0.817i)10-s + (−0.370 − 1.13i)11-s + (−0.889 − 0.646i)12-s + (−0.0482 + 0.148i)13-s + (−0.493 + 0.358i)14-s + (−0.350 + 1.07i)15-s + (−0.379 + 1.16i)16-s + (0.454 − 1.39i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ 0.949 + 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.891298 - 0.142886i\)
\(L(\frac12)\) \(\approx\) \(0.891298 - 0.142886i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (-7.75 - 3.28i)T \)
good2 \( 1 + (0.561 + 1.72i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.906 - 2.79i)T + (-2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.21 - 0.879i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (0.388 + 1.19i)T + (-5.66 + 4.11i)T^{2} \)
11 \( 1 + (1.22 + 3.77i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.173 - 0.535i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.87 + 5.77i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.31 - 7.14i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + 8.89T + 23T^{2} \)
29 \( 1 + (-1.96 - 1.42i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.162 - 0.501i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 - 1.78T + 37T^{2} \)
41 \( 1 - 2.18T + 41T^{2} \)
43 \( 1 + (-0.171 - 0.124i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (3.23 - 9.96i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-0.315 - 0.229i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-5.34 - 3.88i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-0.163 + 0.502i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-2.66 + 1.93i)T + (20.7 - 63.7i)T^{2} \)
73 \( 1 + (-1.58 - 4.86i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.36 + 5.35i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (6.00 + 4.36i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (-7.05 - 5.12i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + 2.53T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.35631138545684003606485649444, −13.83584531572950689454177197415, −11.97998947843556218023511817288, −10.88741411132367002508410326836, −10.03456486952033734549291514701, −9.645740072719567594385361692351, −8.247385377888921915904766004099, −5.81079614846084557811964800299, −3.89985120630841355373482425392, −2.82378970811814943473148230021, 2.23874608400862105064442954111, 5.59792528094066716412172779578, 6.65330785500538280105796049744, 7.65901272219304405330747167741, 8.485206450071030291030709083729, 9.646854508311283885552573501232, 11.92861273731207692747221676773, 12.78451410293404970985737081532, 13.71073483132466301180554653381, 14.80668553715964630051660118517

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