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Properties

Label 2-5-5.2-c34-0-3
Degree $2$
Conductor $5$
Sign $0.334 + 0.942i$
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  + (−1.51e5 − 1.51e5i)2-s + (−1.00e8 + 1.00e8i)3-s + 2.85e10i·4-s + (1.61e11 − 7.45e11i)5-s + 3.02e13·6-s + (−1.52e14 − 1.52e14i)7-s + (1.72e15 − 1.72e15i)8-s − 3.34e15i·9-s + (−1.37e17 + 8.84e16i)10-s − 5.40e17·11-s + (−2.85e18 − 2.85e18i)12-s + (−1.65e18 + 1.65e18i)13-s + 4.61e19i·14-s + (5.85e19 + 9.07e19i)15-s − 3.06e19·16-s + (−2.97e20 − 2.97e20i)17-s + ⋯
L(s)  = 1  + (−1.15 − 1.15i)2-s + (−0.774 + 0.774i)3-s + 1.66i·4-s + (0.211 − 0.977i)5-s + 1.78·6-s + (−0.656 − 0.656i)7-s + (0.765 − 0.765i)8-s − 0.200i·9-s + (−1.37 + 0.884i)10-s − 1.07·11-s + (−1.28 − 1.28i)12-s + (−0.191 + 0.191i)13-s + 1.51i·14-s + (0.593 + 0.920i)15-s − 0.103·16-s + (−0.359 − 0.359i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.334 + 0.942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.334 + 0.942i$
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: \((2,\ 5,\ (\ :17),\ 0.334 + 0.942i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(0.2863168516\)
\(L(\frac12)\) \(\approx\) \(0.2863168516\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.61e11 + 7.45e11i)T \)
good2 \( 1 + (1.51e5 + 1.51e5i)T + 1.71e10iT^{2} \)
3 \( 1 + (1.00e8 - 1.00e8i)T - 1.66e16iT^{2} \)
7 \( 1 + (1.52e14 + 1.52e14i)T + 5.41e28iT^{2} \)
11 \( 1 + 5.40e17T + 2.55e35T^{2} \)
13 \( 1 + (1.65e18 - 1.65e18i)T - 7.48e37iT^{2} \)
17 \( 1 + (2.97e20 + 2.97e20i)T + 6.84e41iT^{2} \)
19 \( 1 - 8.71e21iT - 3.00e43T^{2} \)
23 \( 1 + (1.38e23 - 1.38e23i)T - 1.98e46iT^{2} \)
29 \( 1 - 4.12e24iT - 5.26e49T^{2} \)
31 \( 1 + 2.74e25T + 5.08e50T^{2} \)
37 \( 1 + (-6.11e26 - 6.11e26i)T + 2.08e53iT^{2} \)
41 \( 1 + 9.45e26T + 6.83e54T^{2} \)
43 \( 1 + (-4.03e27 + 4.03e27i)T - 3.45e55iT^{2} \)
47 \( 1 + (3.42e28 + 3.42e28i)T + 7.10e56iT^{2} \)
53 \( 1 + (-1.04e29 + 1.04e29i)T - 4.22e58iT^{2} \)
59 \( 1 + 3.54e29iT - 1.61e60T^{2} \)
61 \( 1 - 2.08e30T + 5.02e60T^{2} \)
67 \( 1 + (3.40e29 + 3.40e29i)T + 1.22e62iT^{2} \)
71 \( 1 + 4.06e30T + 8.76e62T^{2} \)
73 \( 1 + (-1.32e31 + 1.32e31i)T - 2.25e63iT^{2} \)
79 \( 1 - 2.73e32iT - 3.30e64T^{2} \)
83 \( 1 + (-5.18e32 + 5.18e32i)T - 1.77e65iT^{2} \)
89 \( 1 + 5.54e32iT - 1.90e66T^{2} \)
97 \( 1 + (4.46e33 + 4.46e33i)T + 3.55e67iT^{2} \)
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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

−16.19697017931829527551083436479, −13.12701840103770719182518623185, −11.73154363494900159751853744324, −10.32729239125611852598499489909, −9.689171237081892486672578935819, −8.013651074882269597578970473533, −5.41167852685987757466810868158, −3.77472726230806864340708919886, −1.85676148098130836470655274727, −0.35294049243376661080885112020, 0.41295969927243703803800959920, 2.45929927868740592210709788072, 5.78092848013674848241152556644, 6.55518797184770921405794193447, 7.65927421861424421070577966492, 9.420787479616984143222004613721, 10.90539602629765739113826923795, 12.84914169201070675208337803853, 14.95413328994158657547980691583, 16.09188306188718359335489281564

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