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Properties

Label 2-5-5.3-c10-0-2
Degree $2$
Conductor $5$
Sign $0.691 + 0.722i$
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  + (37.4 − 37.4i)2-s + (246. + 246. i)3-s − 1.77e3i·4-s + (−3.02e3 − 784. i)5-s + 1.84e4·6-s + (−8.61e3 + 8.61e3i)7-s + (−2.82e4 − 2.82e4i)8-s + 6.21e4i·9-s + (−1.42e5 + 8.38e4i)10-s + 6.75e4·11-s + (4.38e5 − 4.38e5i)12-s + (8.34e3 + 8.34e3i)13-s + 6.44e5i·14-s + (−5.51e5 − 9.37e5i)15-s − 2.95e5·16-s + (−6.93e5 + 6.93e5i)17-s + ⋯
L(s)  = 1  + (1.16 − 1.16i)2-s + (1.01 + 1.01i)3-s − 1.73i·4-s + (−0.968 − 0.250i)5-s + 2.37·6-s + (−0.512 + 0.512i)7-s + (−0.863 − 0.863i)8-s + 1.05i·9-s + (−1.42 + 0.838i)10-s + 0.419·11-s + (1.76 − 1.76i)12-s + (0.0224 + 0.0224i)13-s + 1.19i·14-s + (−0.726 − 1.23i)15-s − 0.281·16-s + (−0.488 + 0.488i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $0.691 + 0.722i$
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: \((2,\ 5,\ (\ :5),\ 0.691 + 0.722i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.43891 - 1.04155i\)
\(L(\frac12)\) \(\approx\) \(2.43891 - 1.04155i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (3.02e3 + 784. i)T \)
good2 \( 1 + (-37.4 + 37.4i)T - 1.02e3iT^{2} \)
3 \( 1 + (-246. - 246. i)T + 5.90e4iT^{2} \)
7 \( 1 + (8.61e3 - 8.61e3i)T - 2.82e8iT^{2} \)
11 \( 1 - 6.75e4T + 2.59e10T^{2} \)
13 \( 1 + (-8.34e3 - 8.34e3i)T + 1.37e11iT^{2} \)
17 \( 1 + (6.93e5 - 6.93e5i)T - 2.01e12iT^{2} \)
19 \( 1 + 4.48e6iT - 6.13e12T^{2} \)
23 \( 1 + (-1.04e6 - 1.04e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 1.05e7iT - 4.20e14T^{2} \)
31 \( 1 + 2.77e7T + 8.19e14T^{2} \)
37 \( 1 + (-3.89e7 + 3.89e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 7.21e7T + 1.34e16T^{2} \)
43 \( 1 + (-1.47e8 - 1.47e8i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.85e8 - 1.85e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (2.40e8 + 2.40e8i)T + 1.74e17iT^{2} \)
59 \( 1 - 8.16e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.84e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.74e8 + 1.74e8i)T - 1.82e18iT^{2} \)
71 \( 1 + 7.65e8T + 3.25e18T^{2} \)
73 \( 1 + (-1.80e9 - 1.80e9i)T + 4.29e18iT^{2} \)
79 \( 1 + 1.02e9iT - 9.46e18T^{2} \)
83 \( 1 + (3.18e9 + 3.18e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 6.88e9iT - 3.11e19T^{2} \)
97 \( 1 + (-1.10e10 + 1.10e10i)T - 7.37e19iT^{2} \)
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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

−21.42796139593181099454441270256, −20.04639350038412895650628066857, −19.48581149206377270632491509524, −15.73280155441299621426582832881, −14.66171686473800869581779565343, −12.88629587104078139995537882315, −11.16074102083921181592777781839, −9.142027660077886852561714489721, −4.42820064536268404564343537047, −3.04473758620396311089359356956, 3.64886711839483607805681641562, 6.81881278039307277189641259331, 7.992976646016534302038339622368, 12.43803241174790863024068157813, 13.77422666013808210708764469036, 14.87576632599974059497896785272, 16.44127130398696610341081951197, 18.85302551390050767882596900645, 20.23084290471493440536680975800, 22.54420728567791811857610672566

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