↑

Properties

Label 2-10-5.4-c13-0-4
Degree $2$
Conductor $10$
Sign $-0.300 + 0.953i$
Analytic cond. $10.7230$
Root an. cond. $3.27461$
Motivic weight $13$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 64i·2-s + 180. i·3-s − 4.09e3·4-s + (3.33e4 + 1.04e4i)5-s + 1.15e4·6-s − 3.86e5i·7-s + 2.62e5i·8-s + 1.56e6·9-s + (6.71e5 − 2.13e6i)10-s − 8.76e6·11-s − 7.38e5i·12-s − 3.09e7i·13-s − 2.47e7·14-s + (−1.89e6 + 6.00e6i)15-s + 1.67e7·16-s − 1.18e8i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.142i·3-s − 0.5·4-s + (0.953 + 0.300i)5-s + 0.100·6-s − 1.24i·7-s + 0.353i·8-s + 0.979·9-s + (0.212 − 0.674i)10-s − 1.49·11-s − 0.0714i·12-s − 1.77i·13-s − 0.878·14-s + (−0.0428 + 0.136i)15-s + 0.250·16-s − 1.19i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.300 + 0.953i$
Analytic conductor: \(10.7230\)
Root analytic conductor: \(3.27461\)
Motivic weight: \(13\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :13/2),\ -0.300 + 0.953i)\)

Particular Values

\(L(7)\) \(\approx\) \(1.04866 - 1.42966i\)
\(L(\frac12)\) \(\approx\) \(1.04866 - 1.42966i\)
\(L(\frac{15}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 64iT \)
5 \( 1 + (-3.33e4 - 1.04e4i)T \)
good3 \( 1 - 180. iT - 1.59e6T^{2} \)
7 \( 1 + 3.86e5iT - 9.68e10T^{2} \)
11 \( 1 + 8.76e6T + 3.45e13T^{2} \)
13 \( 1 + 3.09e7iT - 3.02e14T^{2} \)
17 \( 1 + 1.18e8iT - 9.90e15T^{2} \)
19 \( 1 - 9.77e7T + 4.20e16T^{2} \)
23 \( 1 - 2.65e8iT - 5.04e17T^{2} \)
29 \( 1 - 3.94e9T + 1.02e19T^{2} \)
31 \( 1 - 4.11e8T + 2.44e19T^{2} \)
37 \( 1 + 4.34e9iT - 2.43e20T^{2} \)
41 \( 1 + 4.06e10T + 9.25e20T^{2} \)
43 \( 1 + 2.14e10iT - 1.71e21T^{2} \)
47 \( 1 - 1.15e11iT - 5.46e21T^{2} \)
53 \( 1 - 1.13e11iT - 2.60e22T^{2} \)
59 \( 1 + 7.68e10T + 1.04e23T^{2} \)
61 \( 1 - 1.14e11T + 1.61e23T^{2} \)
67 \( 1 - 1.23e12iT - 5.48e23T^{2} \)
71 \( 1 - 6.95e11T + 1.16e24T^{2} \)
73 \( 1 + 3.96e11iT - 1.67e24T^{2} \)
79 \( 1 - 5.70e11T + 4.66e24T^{2} \)
83 \( 1 + 1.89e12iT - 8.87e24T^{2} \)
89 \( 1 + 4.80e11T + 2.19e25T^{2} \)
97 \( 1 + 3.52e12iT - 6.73e25T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.59034529098098488892804616429, −15.74406592716377880677574955138, −13.76316255767835613062430001825, −12.94705682707130235333420559588, −10.55391418521005433996944968417, −10.01756725321176748574904647156, −7.51528786160312530224685357477, −5.08868911884824691715741164096, −2.93357209770896062760449975051, −0.862042731152666796046280471896, 1.96512125483128980072436398489, 4.98878442543831491806080866309, 6.48909561852359795711573555200, 8.537414012898196351336020805840, 9.971185789436909102501634881826, 12.44913938356477564050736242793, 13.66709847064020902810497777160, 15.30696340272417674857691913611, 16.47717248740048958398271475605, 18.10219427535810122991533351813

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