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Properties

Label 2-5-5.2-c34-0-15
Degree $2$
Conductor $5$
Sign $0.871 - 0.489i$
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  + (−1.15e5 − 1.15e5i)2-s + (1.39e8 − 1.39e8i)3-s + 9.46e9i·4-s + (−7.62e11 − 3.18e10i)5-s − 3.21e13·6-s + (−2.97e14 − 2.97e14i)7-s + (−8.90e14 + 8.90e14i)8-s − 2.21e16i·9-s + (8.43e16 + 9.16e16i)10-s − 1.99e17·11-s + (1.31e18 + 1.31e18i)12-s + (3.71e18 − 3.71e18i)13-s + 6.87e19i·14-s + (−1.10e20 + 1.01e20i)15-s + 3.68e20·16-s + (−4.39e20 − 4.39e20i)17-s + ⋯
L(s)  = 1  + (−0.880 − 0.880i)2-s + (1.07 − 1.07i)3-s + 0.551i·4-s + (−0.999 − 0.0417i)5-s − 1.90·6-s + (−1.28 − 1.28i)7-s + (−0.395 + 0.395i)8-s − 1.32i·9-s + (0.843 + 0.916i)10-s − 0.394·11-s + (0.594 + 0.594i)12-s + (0.429 − 0.429i)13-s + 2.25i·14-s + (−1.12 + 1.03i)15-s + 1.24·16-s + (−0.531 − 0.531i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.871 - 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.871 - 0.489i$
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: \((2,\ 5,\ (\ :17),\ 0.871 - 0.489i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.4103363169\)
\(L(\frac12)\) \(\approx\) \(0.4103363169\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (7.62e11 + 3.18e10i)T \)
good2 \( 1 + (1.15e5 + 1.15e5i)T + 1.71e10iT^{2} \)
3 \( 1 + (-1.39e8 + 1.39e8i)T - 1.66e16iT^{2} \)
7 \( 1 + (2.97e14 + 2.97e14i)T + 5.41e28iT^{2} \)
11 \( 1 + 1.99e17T + 2.55e35T^{2} \)
13 \( 1 + (-3.71e18 + 3.71e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (4.39e20 + 4.39e20i)T + 6.84e41iT^{2} \)
19 \( 1 + 7.08e21iT - 3.00e43T^{2} \)
23 \( 1 + (1.16e23 - 1.16e23i)T - 1.98e46iT^{2} \)
29 \( 1 + 7.43e23iT - 5.26e49T^{2} \)
31 \( 1 - 1.56e25T + 5.08e50T^{2} \)
37 \( 1 + (4.19e26 + 4.19e26i)T + 2.08e53iT^{2} \)
41 \( 1 - 4.69e27T + 6.83e54T^{2} \)
43 \( 1 + (-1.38e27 + 1.38e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (1.72e28 + 1.72e28i)T + 7.10e56iT^{2} \)
53 \( 1 + (6.65e28 - 6.65e28i)T - 4.22e58iT^{2} \)
59 \( 1 + 1.60e30iT - 1.61e60T^{2} \)
61 \( 1 - 1.31e30T + 5.02e60T^{2} \)
67 \( 1 + (2.97e30 + 2.97e30i)T + 1.22e62iT^{2} \)
71 \( 1 - 3.43e31T + 8.76e62T^{2} \)
73 \( 1 + (-9.17e30 + 9.17e30i)T - 2.25e63iT^{2} \)
79 \( 1 - 8.66e31iT - 3.30e64T^{2} \)
83 \( 1 + (2.22e32 - 2.22e32i)T - 1.77e65iT^{2} \)
89 \( 1 - 1.58e33iT - 1.90e66T^{2} \)
97 \( 1 + (1.71e33 + 1.71e33i)T + 3.55e67iT^{2} \)
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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

−13.76605440451046511856754028206, −12.64766866657173030335065171802, −10.95116258577712668707433319456, −9.390069307505795895710641303079, −8.030403356672993407395917875709, −6.95267044393955006444895099502, −3.58740342600202345615821837073, −2.57773120869037436546279015836, −0.887747576185092956052578850381, −0.18553380279715312871192419732, 2.84711163108888968551928360160, 3.92765634075958360119298736362, 6.26969464419431786664059744160, 8.151894192346983256241516592173, 8.880345334230463821334575219740, 10.03403382600438894226140513828, 12.42726069074927818761645713458, 14.79886006916363941704019359641, 15.83315459501577042436791523984, 16.16957928991885401341090737752

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