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Properties

Label 2-5-5.3-c10-0-0
Degree $2$
Conductor $5$
Sign $0.224 - 0.974i$
Analytic cond. $3.17678$
Root an. cond. $1.78235$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−4.63 + 4.63i)2-s + (70.6 + 70.6i)3-s + 981. i·4-s + (1.00e3 + 2.95e3i)5-s − 654.·6-s + (2.61e3 − 2.61e3i)7-s + (−9.29e3 − 9.29e3i)8-s − 4.90e4i·9-s + (−1.83e4 − 9.05e3i)10-s + 1.56e4·11-s + (−6.93e4 + 6.93e4i)12-s + (2.94e5 + 2.94e5i)13-s + 2.42e4i·14-s + (−1.38e5 + 2.80e5i)15-s − 9.18e5·16-s + (1.23e6 − 1.23e6i)17-s + ⋯
L(s)  = 1  + (−0.144 + 0.144i)2-s + (0.290 + 0.290i)3-s + 0.958i·4-s + (0.321 + 0.946i)5-s − 0.0841·6-s + (0.155 − 0.155i)7-s + (−0.283 − 0.283i)8-s − 0.830i·9-s + (−0.183 − 0.0905i)10-s + 0.0974·11-s + (−0.278 + 0.278i)12-s + (0.792 + 0.792i)13-s + 0.0450i·14-s + (−0.181 + 0.368i)15-s − 0.875·16-s + (0.869 − 0.869i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.224 - 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.224 - 0.974i$
Analytic conductor: \(3.17678\)
Root analytic conductor: \(1.78235\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :5),\ 0.224 - 0.974i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.14089 + 0.908158i\)
\(L(\frac12)\) \(\approx\) \(1.14089 + 0.908158i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.00e3 - 2.95e3i)T \)
good2 \( 1 + (4.63 - 4.63i)T - 1.02e3iT^{2} \)
3 \( 1 + (-70.6 - 70.6i)T + 5.90e4iT^{2} \)
7 \( 1 + (-2.61e3 + 2.61e3i)T - 2.82e8iT^{2} \)
11 \( 1 - 1.56e4T + 2.59e10T^{2} \)
13 \( 1 + (-2.94e5 - 2.94e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-1.23e6 + 1.23e6i)T - 2.01e12iT^{2} \)
19 \( 1 + 2.66e6iT - 6.13e12T^{2} \)
23 \( 1 + (-7.36e6 - 7.36e6i)T + 4.14e13iT^{2} \)
29 \( 1 + 1.66e7iT - 4.20e14T^{2} \)
31 \( 1 + 2.74e7T + 8.19e14T^{2} \)
37 \( 1 + (6.47e7 - 6.47e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 5.65e7T + 1.34e16T^{2} \)
43 \( 1 + (1.85e8 + 1.85e8i)T + 2.16e16iT^{2} \)
47 \( 1 + (-6.69e7 + 6.69e7i)T - 5.25e16iT^{2} \)
53 \( 1 + (-1.86e8 - 1.86e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 7.53e8iT - 5.11e17T^{2} \)
61 \( 1 + 3.22e8T + 7.13e17T^{2} \)
67 \( 1 + (-9.46e8 + 9.46e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 1.70e9T + 3.25e18T^{2} \)
73 \( 1 + (-8.54e8 - 8.54e8i)T + 4.29e18iT^{2} \)
79 \( 1 + 1.15e9iT - 9.46e18T^{2} \)
83 \( 1 + (3.46e9 + 3.46e9i)T + 1.55e19iT^{2} \)
89 \( 1 - 6.69e9iT - 3.11e19T^{2} \)
97 \( 1 + (-1.86e9 + 1.86e9i)T - 7.37e19iT^{2} \)
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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

−21.74862840066785293996085138479, −20.76750065872413375945130909142, −18.58433398629863676030096248783, −17.26968445683209701229566204537, −15.46004246073732419799531269052, −13.73384200556303115267144645141, −11.53977429569763190751209884364, −9.237505785278866123712554016825, −7.01144429536492320766788166836, −3.35724568173085265200334080514, 1.43546105305818356951547383779, 5.48539208092022578871921734289, 8.534007014056794429383521037516, 10.50286472166712082492505018167, 12.89451625360576157420950633364, 14.52923518794813810958837785988, 16.47086543192570213205342773650, 18.40995525372646777303443001698, 19.73087938379554005309595430190, 20.97556093955159651266736587403

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