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Properties

Label 2-10-5.2-c14-0-5
Degree $2$
Conductor $10$
Sign $0.999 - 0.0357i$
Analytic cond. $12.4328$
Root an. cond. $3.52603$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (64 + 64i)2-s + (3.06e3 − 3.06e3i)3-s + 8.19e3i·4-s + (2.06e4 + 7.53e4i)5-s + 3.91e5·6-s + (5.86e5 + 5.86e5i)7-s + (−5.24e5 + 5.24e5i)8-s − 1.39e7i·9-s + (−3.49e6 + 6.14e6i)10-s + 1.03e7·11-s + (2.50e7 + 2.50e7i)12-s + (2.36e7 − 2.36e7i)13-s + 7.50e7i·14-s + (2.93e8 + 1.67e8i)15-s − 6.71e7·16-s + (−2.68e8 − 2.68e8i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (1.39 − 1.39i)3-s + 0.5i·4-s + (0.264 + 0.964i)5-s + 1.39·6-s + (0.711 + 0.711i)7-s + (−0.250 + 0.250i)8-s − 2.91i·9-s + (−0.349 + 0.614i)10-s + 0.531·11-s + (0.699 + 0.699i)12-s + (0.376 − 0.376i)13-s + 0.711i·14-s + (1.71 + 0.979i)15-s − 0.250·16-s + (−0.653 − 0.653i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0357i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.999 - 0.0357i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.999 - 0.0357i$
Analytic conductor: \(12.4328\)
Root analytic conductor: \(3.52603\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :7),\ 0.999 - 0.0357i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(3.88437 + 0.0695276i\)
\(L(\frac12)\) \(\approx\) \(3.88437 + 0.0695276i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-64 - 64i)T \)
5 \( 1 + (-2.06e4 - 7.53e4i)T \)
good3 \( 1 + (-3.06e3 + 3.06e3i)T - 4.78e6iT^{2} \)
7 \( 1 + (-5.86e5 - 5.86e5i)T + 6.78e11iT^{2} \)
11 \( 1 - 1.03e7T + 3.79e14T^{2} \)
13 \( 1 + (-2.36e7 + 2.36e7i)T - 3.93e15iT^{2} \)
17 \( 1 + (2.68e8 + 2.68e8i)T + 1.68e17iT^{2} \)
19 \( 1 - 4.55e8iT - 7.99e17T^{2} \)
23 \( 1 + (3.95e7 - 3.95e7i)T - 1.15e19iT^{2} \)
29 \( 1 - 1.04e10iT - 2.97e20T^{2} \)
31 \( 1 + 1.10e10T + 7.56e20T^{2} \)
37 \( 1 + (7.04e9 + 7.04e9i)T + 9.01e21iT^{2} \)
41 \( 1 + 2.23e10T + 3.79e22T^{2} \)
43 \( 1 + (2.16e11 - 2.16e11i)T - 7.38e22iT^{2} \)
47 \( 1 + (2.30e11 + 2.30e11i)T + 2.56e23iT^{2} \)
53 \( 1 + (3.91e11 - 3.91e11i)T - 1.37e24iT^{2} \)
59 \( 1 - 4.44e12iT - 6.19e24T^{2} \)
61 \( 1 + 2.34e12T + 9.87e24T^{2} \)
67 \( 1 + (5.17e12 + 5.17e12i)T + 3.67e25iT^{2} \)
71 \( 1 - 1.06e13T + 8.27e25T^{2} \)
73 \( 1 + (-7.66e11 + 7.66e11i)T - 1.22e26iT^{2} \)
79 \( 1 + 2.49e13iT - 3.68e26T^{2} \)
83 \( 1 + (-2.02e13 + 2.02e13i)T - 7.36e26iT^{2} \)
89 \( 1 + 5.70e13iT - 1.95e27T^{2} \)
97 \( 1 + (-1.04e14 - 1.04e14i)T + 6.52e27iT^{2} \)
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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

−17.89890983410770865065425035481, −15.12868924290956385340170614573, −14.39416055557850558407962705646, −13.35814152920104400714906818149, −11.85692853294414018969821010435, −8.904115665503826775200664331824, −7.58295278099560669685996530728, −6.32893321261473851469671233511, −3.20820385711959862945903493130, −1.87983804809525247919443860652, 1.84730347520249692770659888793, 3.86545123183683878876312190212, 4.78773722628905976724535508570, 8.392840634234633617818059867675, 9.559114811688637250785244975759, 10.98597650481629184138030515979, 13.37162962425370269565057657651, 14.28277945336072557532676146312, 15.56912958252111676777830038497, 16.94994207245584178461673327876

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