Properties

Label 2-5-5.3-c34-0-7
Degree $2$
Conductor $5$
Sign $0.946 - 0.323i$
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  + (−3.96e4 + 3.96e4i)2-s + (9.37e5 + 9.37e5i)3-s + 1.40e10i·4-s + (−4.84e11 + 5.89e11i)5-s − 7.42e10·6-s + (1.00e14 − 1.00e14i)7-s + (−1.23e15 − 1.23e15i)8-s − 1.66e16i·9-s + (−4.14e15 − 4.25e16i)10-s + 3.11e17·11-s + (−1.31e16 + 1.31e16i)12-s + (−1.45e18 − 1.45e18i)13-s + 7.95e18i·14-s + (−1.00e18 + 9.80e16i)15-s − 1.43e20·16-s + (1.83e20 − 1.83e20i)17-s + ⋯
L(s)  = 1  + (−0.302 + 0.302i)2-s + (0.00725 + 0.00725i)3-s + 0.817i·4-s + (−0.635 + 0.772i)5-s − 0.00438·6-s + (0.431 − 0.431i)7-s + (−0.549 − 0.549i)8-s − 0.999i·9-s + (−0.0414 − 0.425i)10-s + 0.616·11-s + (−0.00592 + 0.00592i)12-s + (−0.168 − 0.168i)13-s + 0.260i·14-s + (−0.0102 + 0.000994i)15-s − 0.484·16-s + (0.221 − 0.221i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(35-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17) \, L(s)\cr =\mathstrut & (0.946 - 0.323i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.946 - 0.323i$
Analytic conductor: \(36.6128\)
Root analytic conductor: \(6.05085\)
Motivic weight: \(34\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17),\ 0.946 - 0.323i)\)

Particular Values

\(L(\frac{35}{2})\) \(\approx\) \(1.419897890\)
\(L(\frac12)\) \(\approx\) \(1.419897890\)
\(L(18)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (4.84e11 - 5.89e11i)T \)
good2 \( 1 + (3.96e4 - 3.96e4i)T - 1.71e10iT^{2} \)
3 \( 1 + (-9.37e5 - 9.37e5i)T + 1.66e16iT^{2} \)
7 \( 1 + (-1.00e14 + 1.00e14i)T - 5.41e28iT^{2} \)
11 \( 1 - 3.11e17T + 2.55e35T^{2} \)
13 \( 1 + (1.45e18 + 1.45e18i)T + 7.48e37iT^{2} \)
17 \( 1 + (-1.83e20 + 1.83e20i)T - 6.84e41iT^{2} \)
19 \( 1 + 4.31e21iT - 3.00e43T^{2} \)
23 \( 1 + (-2.30e22 - 2.30e22i)T + 1.98e46iT^{2} \)
29 \( 1 - 8.56e24iT - 5.26e49T^{2} \)
31 \( 1 - 3.12e24T + 5.08e50T^{2} \)
37 \( 1 + (-9.06e25 + 9.06e25i)T - 2.08e53iT^{2} \)
41 \( 1 - 3.17e27T + 6.83e54T^{2} \)
43 \( 1 + (-6.87e27 - 6.87e27i)T + 3.45e55iT^{2} \)
47 \( 1 + (-3.13e28 + 3.13e28i)T - 7.10e56iT^{2} \)
53 \( 1 + (1.45e29 + 1.45e29i)T + 4.22e58iT^{2} \)
59 \( 1 + 2.55e29iT - 1.61e60T^{2} \)
61 \( 1 + 3.55e30T + 5.02e60T^{2} \)
67 \( 1 + (-2.14e30 + 2.14e30i)T - 1.22e62iT^{2} \)
71 \( 1 - 4.99e31T + 8.76e62T^{2} \)
73 \( 1 + (-4.72e31 - 4.72e31i)T + 2.25e63iT^{2} \)
79 \( 1 + 6.65e31iT - 3.30e64T^{2} \)
83 \( 1 + (1.35e32 + 1.35e32i)T + 1.77e65iT^{2} \)
89 \( 1 + 1.02e33iT - 1.90e66T^{2} \)
97 \( 1 + (-2.80e33 + 2.80e33i)T - 3.55e67iT^{2} \)
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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

−15.70991484268606767758866102351, −14.40685288350879492918620588081, −12.41159666014031853860818091681, −11.18245100039553204126908250373, −9.184213185064485572492318700516, −7.64419793337615328295602427523, −6.64158309504404112428814230602, −4.11227387024979879089904922996, −2.99618274388459238557677508945, −0.63903600683184630801140405976, 0.912178481532823559601957868761, 2.11781817002578749978227381214, 4.42338754674452200937897430576, 5.71740699020993832077403616886, 7.944146668733606465770783470987, 9.269761408715208063648478482732, 10.90325393226057613175888127872, 12.16562787931685557049502264805, 14.09326414007994407955408693951, 15.51098159150578436031034578064

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