Properties

Label 2-71-71.15-c1-0-1
Degree $2$
Conductor $71$
Sign $0.457 - 0.889i$
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  + (−0.569 + 1.33i)2-s + (1.33 + 0.243i)3-s + (−0.0679 − 0.0710i)4-s + (0.862 − 2.65i)5-s + (−1.08 + 1.64i)6-s + (−2.26 + 1.98i)7-s + (−2.57 + 0.967i)8-s + (−1.07 − 0.402i)9-s + (3.04 + 2.66i)10-s + (4.37 − 1.20i)11-s + (−0.0737 − 0.111i)12-s + (−5.09 − 1.40i)13-s + (−1.34 − 4.15i)14-s + (1.80 − 3.34i)15-s + (0.187 − 4.18i)16-s + (1.64 − 1.19i)17-s + ⋯
L(s)  = 1  + (−0.402 + 0.941i)2-s + (0.773 + 0.140i)3-s + (−0.0339 − 0.0355i)4-s + (0.385 − 1.18i)5-s + (−0.443 + 0.671i)6-s + (−0.857 + 0.749i)7-s + (−0.911 + 0.342i)8-s + (−0.357 − 0.134i)9-s + (0.963 + 0.841i)10-s + (1.32 − 0.364i)11-s + (−0.0212 − 0.0322i)12-s + (−1.41 − 0.390i)13-s + (−0.360 − 1.10i)14-s + (0.465 − 0.864i)15-s + (0.0469 − 1.04i)16-s + (0.398 − 0.289i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 71 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.457 - 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(71\)
Sign: $0.457 - 0.889i$
Analytic conductor: \(0.566937\)
Root analytic conductor: \(0.752952\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{71} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 71,\ (\ :1/2),\ 0.457 - 0.889i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.799129 + 0.487398i\)
\(L(\frac12)\) \(\approx\) \(0.799129 + 0.487398i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad71 \( 1 + (-4.00 - 7.41i)T \)
good2 \( 1 + (0.569 - 1.33i)T + (-1.38 - 1.44i)T^{2} \)
3 \( 1 + (-1.33 - 0.243i)T + (2.80 + 1.05i)T^{2} \)
5 \( 1 + (-0.862 + 2.65i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (2.26 - 1.98i)T + (0.939 - 6.93i)T^{2} \)
11 \( 1 + (-4.37 + 1.20i)T + (9.44 - 5.64i)T^{2} \)
13 \( 1 + (5.09 + 1.40i)T + (11.1 + 6.66i)T^{2} \)
17 \( 1 + (-1.64 + 1.19i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.57 - 2.93i)T + (-10.4 + 15.8i)T^{2} \)
23 \( 1 + (-1.08 + 4.75i)T + (-20.7 - 9.97i)T^{2} \)
29 \( 1 + (-1.05 - 7.77i)T + (-27.9 + 7.71i)T^{2} \)
31 \( 1 + (-0.0798 - 1.77i)T + (-30.8 + 2.77i)T^{2} \)
37 \( 1 + (-1.24 - 5.43i)T + (-33.3 + 16.0i)T^{2} \)
41 \( 1 + (2.63 + 3.29i)T + (-9.12 + 39.9i)T^{2} \)
43 \( 1 + (-6.75 - 0.608i)T + (42.3 + 7.67i)T^{2} \)
47 \( 1 + (9.05 - 1.64i)T + (44.0 - 16.5i)T^{2} \)
53 \( 1 + (-0.00506 + 0.00530i)T + (-2.37 - 52.9i)T^{2} \)
59 \( 1 + (2.15 + 3.26i)T + (-23.1 + 54.2i)T^{2} \)
61 \( 1 + (2.44 + 2.13i)T + (8.18 + 60.4i)T^{2} \)
67 \( 1 + (-3.36 - 3.51i)T + (-3.00 + 66.9i)T^{2} \)
73 \( 1 + (-0.0284 + 0.0664i)T + (-50.4 - 52.7i)T^{2} \)
79 \( 1 + (-11.0 + 4.13i)T + (59.4 - 51.9i)T^{2} \)
83 \( 1 + (-2.61 - 3.95i)T + (-32.6 + 76.3i)T^{2} \)
89 \( 1 + (-7.88 + 8.24i)T + (-3.99 - 88.9i)T^{2} \)
97 \( 1 + (11.5 - 14.5i)T + (-21.5 - 94.5i)T^{2} \)
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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

−14.87290205226969924916627786209, −14.23982362396703446988332909061, −12.57034528892460498743221746256, −12.02561285940472504316875405819, −9.546970434790528762955383283705, −9.059202347972993609175397705272, −8.168971592977191115754921022755, −6.59211853227661713877202243241, −5.35820750049783639090855612973, −3.03441496665753651778186429294, 2.35835351724345313939987369306, 3.51365658431700282687931433630, 6.37543746747243414163247255099, 7.36351295365907100075017872449, 9.440277084033136979560944885293, 9.784061806058948778373735236814, 11.05626572911629380163731132079, 12.04452763543725302237210320664, 13.51554798669699934672443122039, 14.45369861605859546323840961709

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