Properties

Label 2-5-5.3-c6-0-1
Degree $2$
Conductor $5$
Sign $0.914 + 0.404i$
Analytic cond. $1.15027$
Root an. cond. $1.07250$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.58 − 4.58i)2-s + (0.411 + 0.411i)3-s + 21.8i·4-s + (−123. + 17.0i)5-s + 3.77·6-s + (215. − 215. i)7-s + (394. + 394. i)8-s − 728. i·9-s + (−489. + 646. i)10-s − 1.32e3·11-s + (−9.00 + 9.00i)12-s + (1.43e3 + 1.43e3i)13-s − 1.97e3i·14-s + (−57.9 − 43.9i)15-s + 2.21e3·16-s + (1.34e3 − 1.34e3i)17-s + ⋯
L(s)  = 1  + (0.573 − 0.573i)2-s + (0.0152 + 0.0152i)3-s + 0.341i·4-s + (−0.990 + 0.136i)5-s + 0.0174·6-s + (0.628 − 0.628i)7-s + (0.769 + 0.769i)8-s − 0.999i·9-s + (−0.489 + 0.646i)10-s − 0.996·11-s + (−0.00520 + 0.00520i)12-s + (0.655 + 0.655i)13-s − 0.720i·14-s + (−0.0171 − 0.0130i)15-s + 0.541·16-s + (0.274 − 0.274i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.914 + 0.404i$
Analytic conductor: \(1.15027\)
Root analytic conductor: \(1.07250\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :3),\ 0.914 + 0.404i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.24028 - 0.262216i\)
\(L(\frac12)\) \(\approx\) \(1.24028 - 0.262216i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (123. - 17.0i)T \)
good2 \( 1 + (-4.58 + 4.58i)T - 64iT^{2} \)
3 \( 1 + (-0.411 - 0.411i)T + 729iT^{2} \)
7 \( 1 + (-215. + 215. i)T - 1.17e5iT^{2} \)
11 \( 1 + 1.32e3T + 1.77e6T^{2} \)
13 \( 1 + (-1.43e3 - 1.43e3i)T + 4.82e6iT^{2} \)
17 \( 1 + (-1.34e3 + 1.34e3i)T - 2.41e7iT^{2} \)
19 \( 1 - 9.19e3iT - 4.70e7T^{2} \)
23 \( 1 + (7.49e3 + 7.49e3i)T + 1.48e8iT^{2} \)
29 \( 1 + 5.43e3iT - 5.94e8T^{2} \)
31 \( 1 + 3.27e4T + 8.87e8T^{2} \)
37 \( 1 + (-5.65e4 + 5.65e4i)T - 2.56e9iT^{2} \)
41 \( 1 - 1.77e4T + 4.75e9T^{2} \)
43 \( 1 + (6.39e3 + 6.39e3i)T + 6.32e9iT^{2} \)
47 \( 1 + (3.52e4 - 3.52e4i)T - 1.07e10iT^{2} \)
53 \( 1 + (6.04e4 + 6.04e4i)T + 2.21e10iT^{2} \)
59 \( 1 - 4.60e4iT - 4.21e10T^{2} \)
61 \( 1 + 7.56e4T + 5.15e10T^{2} \)
67 \( 1 + (-1.54e5 + 1.54e5i)T - 9.04e10iT^{2} \)
71 \( 1 - 1.84e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.01e5 - 3.01e5i)T + 1.51e11iT^{2} \)
79 \( 1 + 1.78e5iT - 2.43e11T^{2} \)
83 \( 1 + (-5.65e5 - 5.65e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 2.49e5iT - 4.96e11T^{2} \)
97 \( 1 + (-3.06e4 + 3.06e4i)T - 8.32e11iT^{2} \)
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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

−23.01672011611124721621304828340, −21.09781237482003825936515479258, −20.31252717827085225860310457320, −18.30517695163686365419746645865, −16.34116157394896496198909899820, −14.36733487052311220638803175645, −12.45522604068505471104036455987, −11.07532374756711977603541735835, −7.88298676318602125850158821797, −3.93006190631750773769310300640, 5.12914450680140621565638995670, 7.897656243489242424922631500684, 10.96914804494141550496753799704, 13.23573897879801250106247432929, 15.09544293120487037724998384683, 16.05188646408227432583105387903, 18.50359540752517275670456818036, 19.91801904767663239097347286980, 21.91988837727605791418011938094, 23.38506031867736604776694607259

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