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Properties

Label 2-5-5.2-c18-0-2
Degree $2$
Conductor $5$
Sign $-0.920 + 0.390i$
Analytic cond. $10.2693$
Root an. cond. $3.20457$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (622. + 622. i)2-s + (−2.43e4 + 2.43e4i)3-s + 5.12e5i·4-s + (1.93e6 + 2.95e5i)5-s − 3.03e7·6-s + (−3.63e6 − 3.63e6i)7-s + (−1.55e8 + 1.55e8i)8-s − 8.02e8i·9-s + (1.01e9 + 1.38e9i)10-s + 1.61e9·11-s + (−1.24e10 − 1.24e10i)12-s + (−6.40e9 + 6.40e9i)13-s − 4.52e9i·14-s + (−5.43e10 + 3.98e10i)15-s − 5.95e10·16-s + (6.97e9 + 6.97e9i)17-s + ⋯
L(s)  = 1  + (1.21 + 1.21i)2-s + (−1.23 + 1.23i)3-s + 1.95i·4-s + (0.988 + 0.151i)5-s − 3.01·6-s + (−0.0900 − 0.0900i)7-s + (−1.16 + 1.16i)8-s − 2.07i·9-s + (1.01 + 1.38i)10-s + 0.683·11-s + (−2.42 − 2.42i)12-s + (−0.603 + 0.603i)13-s − 0.218i·14-s + (−1.41 + 1.03i)15-s − 0.865·16-s + (0.0588 + 0.0588i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.920 + 0.390i$
Analytic conductor: \(10.2693\)
Root analytic conductor: \(3.20457\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :9),\ -0.920 + 0.390i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.468365 - 2.30053i\)
\(L(\frac12)\) \(\approx\) \(0.468365 - 2.30053i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.93e6 - 2.95e5i)T \)
good2 \( 1 + (-622. - 622. i)T + 2.62e5iT^{2} \)
3 \( 1 + (2.43e4 - 2.43e4i)T - 3.87e8iT^{2} \)
7 \( 1 + (3.63e6 + 3.63e6i)T + 1.62e15iT^{2} \)
11 \( 1 - 1.61e9T + 5.55e18T^{2} \)
13 \( 1 + (6.40e9 - 6.40e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (-6.97e9 - 6.97e9i)T + 1.40e22iT^{2} \)
19 \( 1 - 1.05e11iT - 1.04e23T^{2} \)
23 \( 1 + (1.83e12 - 1.83e12i)T - 3.24e24iT^{2} \)
29 \( 1 + 1.51e13iT - 2.10e26T^{2} \)
31 \( 1 - 2.32e13T + 6.99e26T^{2} \)
37 \( 1 + (-1.21e13 - 1.21e13i)T + 1.68e28iT^{2} \)
41 \( 1 + 4.41e14T + 1.07e29T^{2} \)
43 \( 1 + (-1.75e14 + 1.75e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (-1.26e15 - 1.26e15i)T + 1.25e30iT^{2} \)
53 \( 1 + (-1.11e15 + 1.11e15i)T - 1.08e31iT^{2} \)
59 \( 1 - 6.28e15iT - 7.50e31T^{2} \)
61 \( 1 - 5.12e15T + 1.36e32T^{2} \)
67 \( 1 + (3.69e15 + 3.69e15i)T + 7.40e32iT^{2} \)
71 \( 1 - 8.23e16T + 2.10e33T^{2} \)
73 \( 1 + (-2.68e16 + 2.68e16i)T - 3.46e33iT^{2} \)
79 \( 1 - 1.50e17iT - 1.43e34T^{2} \)
83 \( 1 + (6.69e16 - 6.69e16i)T - 3.49e34iT^{2} \)
89 \( 1 + 2.86e17iT - 1.22e35T^{2} \)
97 \( 1 + (-8.46e17 - 8.46e17i)T + 5.77e35iT^{2} \)
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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

−21.22346193264763952984862784644, −17.39527176666818426642392815614, −16.69167688131529785744827588748, −15.35377714405905676814606049751, −13.96549790759763970522604958894, −11.93912756385221374859003475295, −9.835158277701707031710898122343, −6.49236985421366235655003979674, −5.43168907411623031314135258744, −4.06872748054016807208556138487, 0.912515869123774828921262765360, 2.23448877161741421127792535961, 5.11970977704732447922440887640, 6.36735072776573626605835047728, 10.42505468629863427889573841203, 11.97070852508968951483051597429, 12.83680065634401407370710256265, 14.06735462960817040364318243794, 17.14356553244742554604538638695, 18.51861072788437278568026506908

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