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Properties

Label 2-5-5.2-c18-0-1
Degree $2$
Conductor $5$
Sign $0.692 - 0.721i$
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  + (−489. − 489. i)2-s + (9.61e3 − 9.61e3i)3-s + 2.16e5i·4-s + (−1.89e6 + 4.86e5i)5-s − 9.41e6·6-s + (−2.11e7 − 2.11e7i)7-s + (−2.21e7 + 2.21e7i)8-s + 2.02e8i·9-s + (1.16e9 + 6.87e8i)10-s + 3.09e9·11-s + (2.08e9 + 2.08e9i)12-s + (−1.96e9 + 1.96e9i)13-s + 2.06e10i·14-s + (−1.35e10 + 2.28e10i)15-s + 7.85e10·16-s + (−3.54e10 − 3.54e10i)17-s + ⋯
L(s)  = 1  + (−0.955 − 0.955i)2-s + (0.488 − 0.488i)3-s + 0.827i·4-s + (−0.968 + 0.249i)5-s − 0.933·6-s + (−0.523 − 0.523i)7-s + (−0.165 + 0.165i)8-s + 0.522i·9-s + (1.16 + 0.687i)10-s + 1.31·11-s + (0.404 + 0.404i)12-s + (−0.185 + 0.185i)13-s + 0.999i·14-s + (−0.351 + 0.594i)15-s + 1.14·16-s + (−0.298 − 0.298i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.692 - 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.692 - 0.721i$
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: \((2,\ 5,\ (\ :9),\ 0.692 - 0.721i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.371148 + 0.158078i\)
\(L(\frac12)\) \(\approx\) \(0.371148 + 0.158078i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (1.89e6 - 4.86e5i)T \)
good2 \( 1 + (489. + 489. i)T + 2.62e5iT^{2} \)
3 \( 1 + (-9.61e3 + 9.61e3i)T - 3.87e8iT^{2} \)
7 \( 1 + (2.11e7 + 2.11e7i)T + 1.62e15iT^{2} \)
11 \( 1 - 3.09e9T + 5.55e18T^{2} \)
13 \( 1 + (1.96e9 - 1.96e9i)T - 1.12e20iT^{2} \)
17 \( 1 + (3.54e10 + 3.54e10i)T + 1.40e22iT^{2} \)
19 \( 1 - 2.04e11iT - 1.04e23T^{2} \)
23 \( 1 + (1.15e12 - 1.15e12i)T - 3.24e24iT^{2} \)
29 \( 1 - 2.75e13iT - 2.10e26T^{2} \)
31 \( 1 + 3.93e13T + 6.99e26T^{2} \)
37 \( 1 + (1.08e14 + 1.08e14i)T + 1.68e28iT^{2} \)
41 \( 1 + 3.02e14T + 1.07e29T^{2} \)
43 \( 1 + (1.47e14 - 1.47e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (-1.07e15 - 1.07e15i)T + 1.25e30iT^{2} \)
53 \( 1 + (2.74e15 - 2.74e15i)T - 1.08e31iT^{2} \)
59 \( 1 - 1.35e16iT - 7.50e31T^{2} \)
61 \( 1 + 4.67e15T + 1.36e32T^{2} \)
67 \( 1 + (1.01e16 + 1.01e16i)T + 7.40e32iT^{2} \)
71 \( 1 + 1.30e16T + 2.10e33T^{2} \)
73 \( 1 + (-6.62e16 + 6.62e16i)T - 3.46e33iT^{2} \)
79 \( 1 - 7.35e16iT - 1.43e34T^{2} \)
83 \( 1 + (-2.83e16 + 2.83e16i)T - 3.49e34iT^{2} \)
89 \( 1 + 3.63e17iT - 1.22e35T^{2} \)
97 \( 1 + (3.25e17 + 3.25e17i)T + 5.77e35iT^{2} \)
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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

−19.64397341127642392111440136013, −18.44846529083429700637963798756, −16.57887802193756488725317699167, −14.33849835902288361488347842373, −12.23638267028938538039232483364, −10.77875114070703802098106761797, −8.967411706217066381422771569797, −7.32861441759074352130021526213, −3.45718138057487525605103528011, −1.53809498293829674907300479998, 0.25305331787013131043366254007, 3.71902837855600734500635613958, 6.61813785590473510313486071547, 8.456023016243520067974006011789, 9.478831216626911558841507059841, 12.12330721069153196370788641760, 14.93312272911230292272192499966, 15.78029389560207830021056689403, 17.16714829306575544696501046653, 18.91146156160990625214426398327

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