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Properties

Label 2-10-5.3-c16-0-0
Degree $2$
Conductor $10$
Sign $0.0820 - 0.996i$
Analytic cond. $16.2324$
Root an. cond. $4.02895$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−128 + 128i)2-s + (−1.35e3 − 1.35e3i)3-s − 3.27e4i·4-s + (−3.71e5 − 1.20e5i)5-s + 3.46e5·6-s + (2.32e5 − 2.32e5i)7-s + (4.19e6 + 4.19e6i)8-s − 3.93e7i·9-s + (6.29e7 − 3.21e7i)10-s + 2.70e7·11-s + (−4.43e7 + 4.43e7i)12-s + (3.12e8 + 3.12e8i)13-s + 5.94e7i·14-s + (3.39e8 + 6.66e8i)15-s − 1.07e9·16-s + (−4.95e9 + 4.95e9i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−0.206 − 0.206i)3-s − 0.5i·4-s + (−0.951 − 0.308i)5-s + 0.206·6-s + (0.0402 − 0.0402i)7-s + (0.250 + 0.250i)8-s − 0.914i·9-s + (0.629 − 0.321i)10-s + 0.126·11-s + (−0.103 + 0.103i)12-s + (0.383 + 0.383i)13-s + 0.0402i·14-s + (0.132 + 0.260i)15-s − 0.250·16-s + (−0.710 + 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0820 - 0.996i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (0.0820 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.0820 - 0.996i$
Analytic conductor: \(16.2324\)
Root analytic conductor: \(4.02895\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :8),\ 0.0820 - 0.996i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.565438 + 0.520818i\)
\(L(\frac12)\) \(\approx\) \(0.565438 + 0.520818i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (128 - 128i)T \)
5 \( 1 + (3.71e5 + 1.20e5i)T \)
good3 \( 1 + (1.35e3 + 1.35e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (-2.32e5 + 2.32e5i)T - 3.32e13iT^{2} \)
11 \( 1 - 2.70e7T + 4.59e16T^{2} \)
13 \( 1 + (-3.12e8 - 3.12e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (4.95e9 - 4.95e9i)T - 4.86e19iT^{2} \)
19 \( 1 - 2.77e10iT - 2.88e20T^{2} \)
23 \( 1 + (-4.85e10 - 4.85e10i)T + 6.13e21iT^{2} \)
29 \( 1 + 1.69e11iT - 2.50e23T^{2} \)
31 \( 1 + 8.83e10T + 7.27e23T^{2} \)
37 \( 1 + (-3.82e12 + 3.82e12i)T - 1.23e25iT^{2} \)
41 \( 1 - 1.29e12T + 6.37e25T^{2} \)
43 \( 1 + (-1.34e13 - 1.34e13i)T + 1.36e26iT^{2} \)
47 \( 1 + (5.93e12 - 5.93e12i)T - 5.66e26iT^{2} \)
53 \( 1 + (-4.58e13 - 4.58e13i)T + 3.87e27iT^{2} \)
59 \( 1 + 2.43e14iT - 2.15e28T^{2} \)
61 \( 1 + 2.08e14T + 3.67e28T^{2} \)
67 \( 1 + (-3.69e13 + 3.69e13i)T - 1.64e29iT^{2} \)
71 \( 1 + 4.12e14T + 4.16e29T^{2} \)
73 \( 1 + (-1.10e14 - 1.10e14i)T + 6.50e29iT^{2} \)
79 \( 1 - 2.00e15iT - 2.30e30T^{2} \)
83 \( 1 + (-1.15e13 - 1.15e13i)T + 5.07e30iT^{2} \)
89 \( 1 - 2.86e15iT - 1.54e31T^{2} \)
97 \( 1 + (-3.85e15 + 3.85e15i)T - 6.14e31iT^{2} \)
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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

−17.10420171274682636443871982075, −15.84581304677309260897982614063, −14.64468934492145064671913758808, −12.58586486181061641759123203580, −11.17117991961257066686914203218, −9.182138868323591957877368351509, −7.74773826627193129013426358934, −6.16236011962404935244098319134, −3.98554475440762464818319114606, −1.12609991174402523715159833705, 0.44584724890944152811742597143, 2.72494373198104791124867427940, 4.59295647140129265805009030447, 7.19827535007125751255471987556, 8.723191551439559380026239262884, 10.66665136266459901287770417898, 11.56114422562382144551983880075, 13.31777745675409502403987169563, 15.34067322548227210959498871796, 16.51114834187803657688005577065

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