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Properties

Label 2-10-5.2-c12-0-0
Degree $2$
Conductor $10$
Sign $-0.839 + 0.542i$
Analytic cond. $9.13993$
Root an. cond. $3.02323$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (32 + 32i)2-s + (−748. + 748. i)3-s + 2.04e3i·4-s + (1.12e4 + 1.08e4i)5-s − 4.79e4·6-s + (−1.21e5 − 1.21e5i)7-s + (−6.55e4 + 6.55e4i)8-s − 5.89e5i·9-s + (1.42e4 + 7.06e5i)10-s − 8.55e5·11-s + (−1.53e6 − 1.53e6i)12-s + (5.63e5 − 5.63e5i)13-s − 7.74e6i·14-s + (−1.65e7 + 3.32e5i)15-s − 4.19e6·16-s + (−1.33e7 − 1.33e7i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−1.02 + 1.02i)3-s + 0.5i·4-s + (0.721 + 0.692i)5-s − 1.02·6-s + (−1.02 − 1.02i)7-s + (−0.250 + 0.250i)8-s − 1.10i·9-s + (0.0142 + 0.706i)10-s − 0.482·11-s + (−0.513 − 0.513i)12-s + (0.116 − 0.116i)13-s − 1.02i·14-s + (−1.45 + 0.0291i)15-s − 0.250·16-s + (−0.554 − 0.554i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.839 + 0.542i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.839 + 0.542i$
Analytic conductor: \(9.13993\)
Root analytic conductor: \(3.02323\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :6),\ -0.839 + 0.542i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.225296 - 0.763812i\)
\(L(\frac12)\) \(\approx\) \(0.225296 - 0.763812i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-32 - 32i)T \)
5 \( 1 + (-1.12e4 - 1.08e4i)T \)
good3 \( 1 + (748. - 748. i)T - 5.31e5iT^{2} \)
7 \( 1 + (1.21e5 + 1.21e5i)T + 1.38e10iT^{2} \)
11 \( 1 + 8.55e5T + 3.13e12T^{2} \)
13 \( 1 + (-5.63e5 + 5.63e5i)T - 2.32e13iT^{2} \)
17 \( 1 + (1.33e7 + 1.33e7i)T + 5.82e14iT^{2} \)
19 \( 1 - 9.07e7iT - 2.21e15T^{2} \)
23 \( 1 + (-6.22e7 + 6.22e7i)T - 2.19e16iT^{2} \)
29 \( 1 - 9.61e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.55e9T + 7.87e17T^{2} \)
37 \( 1 + (-7.41e8 - 7.41e8i)T + 6.58e18iT^{2} \)
41 \( 1 - 9.28e8T + 2.25e19T^{2} \)
43 \( 1 + (2.23e9 - 2.23e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (7.22e9 + 7.22e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (9.05e8 - 9.05e8i)T - 4.91e20iT^{2} \)
59 \( 1 - 1.35e10iT - 1.77e21T^{2} \)
61 \( 1 - 6.94e10T + 2.65e21T^{2} \)
67 \( 1 + (-1.13e11 - 1.13e11i)T + 8.18e21iT^{2} \)
71 \( 1 - 1.09e11T + 1.64e22T^{2} \)
73 \( 1 + (8.58e9 - 8.58e9i)T - 2.29e22iT^{2} \)
79 \( 1 + 1.74e11iT - 5.90e22T^{2} \)
83 \( 1 + (2.87e11 - 2.87e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 2.51e11iT - 2.46e23T^{2} \)
97 \( 1 + (-2.97e11 - 2.97e11i)T + 6.93e23iT^{2} \)
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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

−18.25506611758462402667732245666, −16.78798473807969513034127501442, −16.15735435625745019380751043965, −14.50414797895474068117686798416, −12.98115338024413290269792619982, −10.87104453965064521289882865895, −9.891799756266394524498067565890, −6.83245730931968035585536954929, −5.47244395565717974744352970712, −3.62085323652530120289587286706, 0.35585329146369567201205945431, 2.20718840248673584202608893524, 5.36243544676323038304071291283, 6.46407882442153452334961908703, 9.296944626994536732323096380425, 11.34054190477351581942429078481, 12.77553475684878669751467186822, 13.18545167385466839204771218324, 15.67500072281830242051865709903, 17.31030379748921722201208565443

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