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Properties

Label 2-5-5.2-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: $\chi_{5} (2, \cdot )$
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

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

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