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Properties

Label 2-5-5.2-c20-0-0
Degree $2$
Conductor $5$
Sign $-0.0112 - 0.999i$
Analytic cond. $12.6756$
Root an. cond. $3.56029$
Motivic weight $20$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−439. − 439. i)2-s + (−4.82e4 + 4.82e4i)3-s − 6.61e5i·4-s + (5.04e6 − 8.36e6i)5-s + 4.24e7·6-s + (−2.17e8 − 2.17e8i)7-s + (−7.52e8 + 7.52e8i)8-s − 1.16e9i·9-s + (−5.89e9 + 1.46e9i)10-s + 2.92e10·11-s + (3.19e10 + 3.19e10i)12-s + (−1.32e11 + 1.32e11i)13-s + 1.91e11i·14-s + (1.60e11 + 6.46e11i)15-s − 3.22e10·16-s + (−1.20e11 − 1.20e11i)17-s + ⋯
L(s)  = 1  + (−0.429 − 0.429i)2-s + (−0.816 + 0.816i)3-s − 0.631i·4-s + (0.516 − 0.856i)5-s + 0.701·6-s + (−0.769 − 0.769i)7-s + (−0.700 + 0.700i)8-s − 0.334i·9-s + (−0.589 + 0.146i)10-s + 1.12·11-s + (0.515 + 0.515i)12-s + (−0.963 + 0.963i)13-s + 0.661i·14-s + (0.278 + 1.12i)15-s − 0.0293·16-s + (−0.0600 − 0.0600i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0112 - 0.999i)\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & (-0.0112 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.0112 - 0.999i$
Analytic conductor: \(12.6756\)
Root analytic conductor: \(3.56029\)
Motivic weight: \(20\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :10),\ -0.0112 - 0.999i)\)

Particular Values

\(L(\frac{21}{2})\) \(\approx\) \(0.280733 + 0.283900i\)
\(L(\frac12)\) \(\approx\) \(0.280733 + 0.283900i\)
\(L(11)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-5.04e6 + 8.36e6i)T \)
good2 \( 1 + (439. + 439. i)T + 1.04e6iT^{2} \)
3 \( 1 + (4.82e4 - 4.82e4i)T - 3.48e9iT^{2} \)
7 \( 1 + (2.17e8 + 2.17e8i)T + 7.97e16iT^{2} \)
11 \( 1 - 2.92e10T + 6.72e20T^{2} \)
13 \( 1 + (1.32e11 - 1.32e11i)T - 1.90e22iT^{2} \)
17 \( 1 + (1.20e11 + 1.20e11i)T + 4.06e24iT^{2} \)
19 \( 1 - 1.13e13iT - 3.75e25T^{2} \)
23 \( 1 + (-2.06e11 + 2.06e11i)T - 1.71e27iT^{2} \)
29 \( 1 - 3.26e14iT - 1.76e29T^{2} \)
31 \( 1 - 5.37e14T + 6.71e29T^{2} \)
37 \( 1 + (-2.32e15 - 2.32e15i)T + 2.31e31iT^{2} \)
41 \( 1 - 6.93e14T + 1.80e32T^{2} \)
43 \( 1 + (2.46e15 - 2.46e15i)T - 4.67e32iT^{2} \)
47 \( 1 + (-2.52e16 - 2.52e16i)T + 2.76e33iT^{2} \)
53 \( 1 + (2.12e17 - 2.12e17i)T - 3.05e34iT^{2} \)
59 \( 1 + 1.07e17iT - 2.61e35T^{2} \)
61 \( 1 + 4.78e17T + 5.08e35T^{2} \)
67 \( 1 + (9.30e16 + 9.30e16i)T + 3.32e36iT^{2} \)
71 \( 1 + 4.63e18T + 1.05e37T^{2} \)
73 \( 1 + (4.17e18 - 4.17e18i)T - 1.84e37iT^{2} \)
79 \( 1 + 1.74e19iT - 8.96e37T^{2} \)
83 \( 1 + (6.68e18 - 6.68e18i)T - 2.40e38iT^{2} \)
89 \( 1 - 2.68e19iT - 9.72e38T^{2} \)
97 \( 1 + (-2.43e18 - 2.43e18i)T + 5.43e39iT^{2} \)
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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

−19.25776921966233392033226714869, −17.14772739374778726802640366265, −16.40777562822619627003776131600, −14.21389565228920579794653481960, −11.96650824845364455561126717925, −10.24483556666427192797516378854, −9.408971306078520622800685826713, −6.09114624176780731687042451580, −4.46586952323415050721983598089, −1.39058026553693245579315227640, 0.22892235106816387724810329527, 2.80260063120540767602143525210, 6.16455789266526501785844401067, 7.11343966473767885348342179521, 9.386015867834349997799983424586, 11.73796297559510004725818145812, 12.97413308644183738864842984160, 15.28919663069123242543275919480, 17.22422164341537279882694018303, 17.86990263097775481103617305433

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