Properties

Label 2-5-5.2-c34-0-6
Degree $2$
Conductor $5$
Sign $-0.995 + 0.0926i$
Analytic cond. $36.6128$
Root an. cond. $6.05085$
Motivic weight $34$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.67e5 + 1.67e5i)2-s + (4.67e7 − 4.67e7i)3-s + 3.87e10i·4-s + (6.83e11 + 3.39e11i)5-s + 1.56e13·6-s + (−2.57e14 − 2.57e14i)7-s + (−3.60e15 + 3.60e15i)8-s + 1.23e16i·9-s + (5.75e16 + 1.70e17i)10-s − 4.35e17·11-s + (1.80e18 + 1.80e18i)12-s + (−9.43e18 + 9.43e18i)13-s − 8.59e19i·14-s + (4.77e19 − 1.60e19i)15-s − 5.39e20·16-s + (6.46e20 + 6.46e20i)17-s + ⋯
L(s)  = 1  + (1.27 + 1.27i)2-s + (0.361 − 0.361i)3-s + 2.25i·4-s + (0.895 + 0.444i)5-s + 0.923·6-s + (−1.10 − 1.10i)7-s + (−1.59 + 1.59i)8-s + 0.738i·9-s + (0.575 + 1.70i)10-s − 0.861·11-s + (0.815 + 0.815i)12-s + (−1.09 + 1.09i)13-s − 2.82i·14-s + (0.485 − 0.163i)15-s − 1.82·16-s + (0.781 + 0.781i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0926i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (-0.995 + 0.0926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.995 + 0.0926i$
Analytic conductor: \(36.6128\)
Root analytic conductor: \(6.05085\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17),\ -0.995 + 0.0926i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(3.549180440\)
\(L(\frac12)\) \(\approx\) \(3.549180440\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-6.83e11 - 3.39e11i)T \)
good2 \( 1 + (-1.67e5 - 1.67e5i)T + 1.71e10iT^{2} \)
3 \( 1 + (-4.67e7 + 4.67e7i)T - 1.66e16iT^{2} \)
7 \( 1 + (2.57e14 + 2.57e14i)T + 5.41e28iT^{2} \)
11 \( 1 + 4.35e17T + 2.55e35T^{2} \)
13 \( 1 + (9.43e18 - 9.43e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (-6.46e20 - 6.46e20i)T + 6.84e41iT^{2} \)
19 \( 1 - 3.00e21iT - 3.00e43T^{2} \)
23 \( 1 + (6.72e22 - 6.72e22i)T - 1.98e46iT^{2} \)
29 \( 1 + 5.44e24iT - 5.26e49T^{2} \)
31 \( 1 + 2.33e24T + 5.08e50T^{2} \)
37 \( 1 + (1.67e26 + 1.67e26i)T + 2.08e53iT^{2} \)
41 \( 1 - 2.76e27T + 6.83e54T^{2} \)
43 \( 1 + (-4.51e27 + 4.51e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (1.66e28 + 1.66e28i)T + 7.10e56iT^{2} \)
53 \( 1 + (-1.68e29 + 1.68e29i)T - 4.22e58iT^{2} \)
59 \( 1 + 1.90e29iT - 1.61e60T^{2} \)
61 \( 1 - 1.01e30T + 5.02e60T^{2} \)
67 \( 1 + (-1.07e31 - 1.07e31i)T + 1.22e62iT^{2} \)
71 \( 1 - 1.88e31T + 8.76e62T^{2} \)
73 \( 1 + (1.56e31 - 1.56e31i)T - 2.25e63iT^{2} \)
79 \( 1 - 1.83e31iT - 3.30e64T^{2} \)
83 \( 1 + (2.44e31 - 2.44e31i)T - 1.77e65iT^{2} \)
89 \( 1 - 2.09e33iT - 1.90e66T^{2} \)
97 \( 1 + (-5.32e33 - 5.32e33i)T + 3.55e67iT^{2} \)
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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

−16.25639513742823773868975793588, −14.43413807993224750317629445677, −13.66182094204762019095706322269, −12.75808010779600736002526119357, −10.07701959937657701322156110196, −7.68182575455529922380529492162, −6.78755984504337116793253537104, −5.46758688841921138984270275512, −3.82122353924453599656093452841, −2.36514800525904754698677455806, 0.59166542157628143237326723652, 2.53990667441270674604314861592, 3.02970119956242194955330597862, 4.98231905736888898979713886934, 5.93789105820175744465497902509, 9.364734645208734645737312474047, 10.15613078414426385430598031276, 12.31140521625578730726670887439, 12.88062596108834097965205128764, 14.43370080075157072114451958300

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