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Properties

Label 2-5-5.4-c33-0-7
Degree $2$
Conductor $5$
Sign $-0.964 + 0.263i$
Analytic cond. $34.4914$
Root an. cond. $5.87293$
Motivic weight $33$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.62e5i·2-s + 1.50e7i·3-s − 1.76e10·4-s + (3.29e11 − 8.98e10i)5-s − 2.43e12·6-s + 1.33e14i·7-s − 1.46e15i·8-s + 5.33e15·9-s + (1.45e16 + 5.33e16i)10-s + 2.82e17·11-s − 2.65e17i·12-s + 1.43e18i·13-s − 2.16e19·14-s + (1.35e18 + 4.95e18i)15-s + 8.63e19·16-s + 2.38e20i·17-s + ⋯
L(s)  = 1  + 1.74i·2-s + 0.201i·3-s − 2.05·4-s + (0.964 − 0.263i)5-s − 0.353·6-s + 1.52i·7-s − 1.84i·8-s + 0.959·9-s + (0.460 + 1.68i)10-s + 1.85·11-s − 0.415i·12-s + 0.597i·13-s − 2.65·14-s + (0.0531 + 0.194i)15-s + 1.16·16-s + 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(34-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+33/2) \, L(s)\cr =\mathstrut & (-0.964 + 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.964 + 0.263i$
Analytic conductor: \(34.4914\)
Root analytic conductor: \(5.87293\)
Motivic weight: \(33\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :33/2),\ -0.964 + 0.263i)\)

Particular Values

\(L(17)\) \(\approx\) \(2.581438583\)
\(L(\frac12)\) \(\approx\) \(2.581438583\)
\(L(\frac{35}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.29e11 + 8.98e10i)T \)
good2 \( 1 - 1.62e5iT - 8.58e9T^{2} \)
3 \( 1 - 1.50e7iT - 5.55e15T^{2} \)
7 \( 1 - 1.33e14iT - 7.73e27T^{2} \)
11 \( 1 - 2.82e17T + 2.32e34T^{2} \)
13 \( 1 - 1.43e18iT - 5.75e36T^{2} \)
17 \( 1 - 2.38e20iT - 4.02e40T^{2} \)
19 \( 1 - 5.86e20T + 1.58e42T^{2} \)
23 \( 1 + 1.73e22iT - 8.65e44T^{2} \)
29 \( 1 - 1.13e23T + 1.81e48T^{2} \)
31 \( 1 - 2.12e24T + 1.64e49T^{2} \)
37 \( 1 + 8.11e25iT - 5.63e51T^{2} \)
41 \( 1 + 1.96e26T + 1.66e53T^{2} \)
43 \( 1 + 8.68e26iT - 8.02e53T^{2} \)
47 \( 1 - 5.03e27iT - 1.51e55T^{2} \)
53 \( 1 + 7.43e27iT - 7.96e56T^{2} \)
59 \( 1 + 5.72e28T + 2.74e58T^{2} \)
61 \( 1 + 2.92e29T + 8.23e58T^{2} \)
67 \( 1 + 8.52e29iT - 1.82e60T^{2} \)
71 \( 1 + 1.52e30T + 1.23e61T^{2} \)
73 \( 1 - 1.66e30iT - 3.08e61T^{2} \)
79 \( 1 + 2.86e31T + 4.18e62T^{2} \)
83 \( 1 - 1.27e30iT - 2.13e63T^{2} \)
89 \( 1 + 1.21e32T + 2.13e64T^{2} \)
97 \( 1 - 9.03e32iT - 3.65e65T^{2} \)
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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

−16.47323460333464514231748483907, −15.17546329212989343042546598612, −14.13986545814538273594147764475, −12.45716421224327714059608858291, −9.470170980016046339345781122750, −8.730114404489750182419632851438, −6.65866560135154645074013158884, −5.80449837193242419448922650413, −4.35945060134494368332653988809, −1.61433994871576215816338863089, 0.932528984643437189166860512744, 1.47719122447255762041850712147, 3.25106867355082886647647772614, 4.47971431799989551776588602192, 6.93350376525499830633580690502, 9.495403653247396427277065237047, 10.26299993070367322940930015992, 11.68934736250780184773711774259, 13.29188445702231489002052584724, 14.02522795916019369713258479970

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