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Properties

Label 2-10-5.4-c7-0-1
Degree $2$
Conductor $10$
Sign $0.526 - 0.850i$
Analytic cond. $3.12385$
Root an. cond. $1.76744$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s + 83.6i·3-s − 64·4-s + (237. + 147. i)5-s + 669.·6-s + 185. i·7-s + 512i·8-s − 4.81e3·9-s + (1.17e3 − 1.90e3i)10-s + 3.56e3·11-s − 5.35e3i·12-s − 6.09e3i·13-s + 1.48e3·14-s + (−1.23e4 + 1.98e4i)15-s + 4.09e3·16-s − 1.24e4i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 1.78i·3-s − 0.5·4-s + (0.850 + 0.526i)5-s + 1.26·6-s + 0.204i·7-s + 0.353i·8-s − 2.20·9-s + (0.371 − 0.601i)10-s + 0.806·11-s − 0.894i·12-s − 0.769i·13-s + 0.144·14-s + (−0.941 + 1.52i)15-s + 0.250·16-s − 0.615i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.526 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.526 - 0.850i$
Analytic conductor: \(3.12385\)
Root analytic conductor: \(1.76744\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :7/2),\ 0.526 - 0.850i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.25469 + 0.699239i\)
\(L(\frac12)\) \(\approx\) \(1.25469 + 0.699239i\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
5 \( 1 + (-237. - 147. i)T \)
good3 \( 1 - 83.6iT - 2.18e3T^{2} \)
7 \( 1 - 185. iT - 8.23e5T^{2} \)
11 \( 1 - 3.56e3T + 1.94e7T^{2} \)
13 \( 1 + 6.09e3iT - 6.27e7T^{2} \)
17 \( 1 + 1.24e4iT - 4.10e8T^{2} \)
19 \( 1 - 5.06e4T + 8.93e8T^{2} \)
23 \( 1 + 1.14e4iT - 3.40e9T^{2} \)
29 \( 1 + 1.01e5T + 1.72e10T^{2} \)
31 \( 1 + 2.90e4T + 2.75e10T^{2} \)
37 \( 1 - 1.49e5iT - 9.49e10T^{2} \)
41 \( 1 + 3.74e5T + 1.94e11T^{2} \)
43 \( 1 - 1.74e5iT - 2.71e11T^{2} \)
47 \( 1 + 4.28e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.71e6iT - 1.17e12T^{2} \)
59 \( 1 + 1.34e5T + 2.48e12T^{2} \)
61 \( 1 + 1.39e6T + 3.14e12T^{2} \)
67 \( 1 + 2.60e6iT - 6.06e12T^{2} \)
71 \( 1 + 4.91e6T + 9.09e12T^{2} \)
73 \( 1 - 1.19e5iT - 1.10e13T^{2} \)
79 \( 1 - 4.70e6T + 1.92e13T^{2} \)
83 \( 1 - 9.19e6iT - 2.71e13T^{2} \)
89 \( 1 + 6.43e6T + 4.42e13T^{2} \)
97 \( 1 + 1.26e7iT - 8.07e13T^{2} \)
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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

−20.08935962370004503624185608059, −18.09722229201436429479026567578, −16.71289280521128786009242443137, −15.13451449272596670963511007349, −13.92559472015997342315228999571, −11.47773087953961666413120642771, −10.14295639502063941940733858177, −9.249043236612006197813301497945, −5.32554859408978323330465615091, −3.25971690423980935490263412487, 1.37003475765328559695814707233, 5.93088416668753855450962986916, 7.30979794387154382333110495472, 9.039626562876661235970843121495, 12.04272751746887657229160282832, 13.40234642670442655734536262262, 14.21426954094864078302826728911, 16.73536492962310677015722275597, 17.65843574979804520094317654841, 18.76133535732387945636910078652

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