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Properties

Label 2-71-71.20-c1-0-3
Degree $2$
Conductor $71$
Sign $-0.226 + 0.974i$
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  + (−0.469 − 2.05i)2-s + (1.38 − 0.665i)3-s + (−2.20 + 1.06i)4-s + 0.419·5-s + (−2.01 − 2.53i)6-s + (−0.760 + 3.33i)7-s + (0.594 + 0.745i)8-s + (−0.401 + 0.504i)9-s + (−0.197 − 0.863i)10-s + (0.0892 − 0.111i)11-s + (−2.34 + 2.94i)12-s + (−1.79 − 2.25i)13-s + 7.21·14-s + (0.580 − 0.279i)15-s + (−1.80 + 2.26i)16-s + 3.91·17-s + ⋯
L(s)  = 1  + (−0.331 − 1.45i)2-s + (0.798 − 0.384i)3-s + (−1.10 + 0.531i)4-s + 0.187·5-s + (−0.824 − 1.03i)6-s + (−0.287 + 1.25i)7-s + (0.210 + 0.263i)8-s + (−0.133 + 0.168i)9-s + (−0.0623 − 0.273i)10-s + (0.0269 − 0.0337i)11-s + (−0.677 + 0.849i)12-s + (−0.499 − 0.625i)13-s + 1.92·14-s + (0.149 − 0.0722i)15-s + (−0.450 + 0.565i)16-s + 0.950·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.226 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $-0.226 + 0.974i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (20, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ -0.226 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.578156 - 0.727915i\)
\(L(\frac12)\) \(\approx\) \(0.578156 - 0.727915i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (3.50 + 7.66i)T \)
good2 \( 1 + (0.469 + 2.05i)T + (-1.80 + 0.867i)T^{2} \)
3 \( 1 + (-1.38 + 0.665i)T + (1.87 - 2.34i)T^{2} \)
5 \( 1 - 0.419T + 5T^{2} \)
7 \( 1 + (0.760 - 3.33i)T + (-6.30 - 3.03i)T^{2} \)
11 \( 1 + (-0.0892 + 0.111i)T + (-2.44 - 10.7i)T^{2} \)
13 \( 1 + (1.79 + 2.25i)T + (-2.89 + 12.6i)T^{2} \)
17 \( 1 - 3.91T + 17T^{2} \)
19 \( 1 + (-4.50 - 2.17i)T + (11.8 + 14.8i)T^{2} \)
23 \( 1 + (-1.81 + 7.96i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (6.75 - 3.25i)T + (18.0 - 22.6i)T^{2} \)
31 \( 1 + (3.80 + 4.76i)T + (-6.89 + 30.2i)T^{2} \)
37 \( 1 + (0.874 + 3.83i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (2.22 + 2.79i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-1.86 - 8.19i)T + (-38.7 + 18.6i)T^{2} \)
47 \( 1 + (-5.84 - 2.81i)T + (29.3 + 36.7i)T^{2} \)
53 \( 1 + (4.51 + 2.17i)T + (33.0 + 41.4i)T^{2} \)
59 \( 1 + (1.54 - 1.93i)T + (-13.1 - 57.5i)T^{2} \)
61 \( 1 + (-0.668 - 2.92i)T + (-54.9 + 26.4i)T^{2} \)
67 \( 1 + (-11.0 + 5.32i)T + (41.7 - 52.3i)T^{2} \)
73 \( 1 + (1.19 + 5.24i)T + (-65.7 + 31.6i)T^{2} \)
79 \( 1 + (-10.8 - 13.5i)T + (-17.5 + 77.0i)T^{2} \)
83 \( 1 + (-2.24 + 2.81i)T + (-18.4 - 80.9i)T^{2} \)
89 \( 1 + (8.90 + 4.28i)T + (55.4 + 69.5i)T^{2} \)
97 \( 1 + (-2.14 + 2.68i)T + (-21.5 - 94.5i)T^{2} \)
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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

−14.14614921004204296706016392202, −12.82769348802620229905022942341, −12.30257436943105298047529194922, −11.08314604785265534502939943371, −9.775851230405599622526709572878, −8.989083359584597622041155427032, −7.79370233748159174987863019427, −5.63594429914650982682547108164, −3.23440561590688160482212064501, −2.18613386487835846364881151331, 3.64601277168255406099853554172, 5.46781546176055201337490532363, 7.05155857127583937358691693299, 7.76396820359429771675796722278, 9.233678065660999189828019220970, 9.827516554616635818346202319279, 11.64888983721320000331086290507, 13.62428715105210086628430470301, 14.08024668540860348654886816730, 15.09791895073539551255706009998

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