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Properties

Label 2-10-5.3-c6-0-0
Degree $2$
Conductor $10$
Sign $0.976 + 0.214i$
Analytic cond. $2.30054$
Root an. cond. $1.51675$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4 − 4i)2-s + (23.8 + 23.8i)3-s − 32i·4-s + (54.1 − 112. i)5-s + 191.·6-s + (−362. + 362. i)7-s + (−128 − 128i)8-s + 412. i·9-s + (−234. − 667. i)10-s − 1.32e3·11-s + (764. − 764. i)12-s + (254. + 254. i)13-s + 2.90e3i·14-s + (3.98e3 − 1.39e3i)15-s − 1.02e3·16-s + (4.39e3 − 4.39e3i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.884 + 0.884i)3-s − 0.5i·4-s + (0.432 − 0.901i)5-s + 0.884·6-s + (−1.05 + 1.05i)7-s + (−0.250 − 0.250i)8-s + 0.566i·9-s + (−0.234 − 0.667i)10-s − 0.992·11-s + (0.442 − 0.442i)12-s + (0.115 + 0.115i)13-s + 1.05i·14-s + (1.18 − 0.414i)15-s − 0.250·16-s + (0.895 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.976 + 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.976 + 0.214i$
Analytic conductor: \(2.30054\)
Root analytic conductor: \(1.51675\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :3),\ 0.976 + 0.214i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.90635 - 0.206517i\)
\(L(\frac12)\) \(\approx\) \(1.90635 - 0.206517i\)
\(L(4)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-4 + 4i)T \)
5 \( 1 + (-54.1 + 112. i)T \)
good3 \( 1 + (-23.8 - 23.8i)T + 729iT^{2} \)
7 \( 1 + (362. - 362. i)T - 1.17e5iT^{2} \)
11 \( 1 + 1.32e3T + 1.77e6T^{2} \)
13 \( 1 + (-254. - 254. i)T + 4.82e6iT^{2} \)
17 \( 1 + (-4.39e3 + 4.39e3i)T - 2.41e7iT^{2} \)
19 \( 1 - 819. iT - 4.70e7T^{2} \)
23 \( 1 + (-1.11e4 - 1.11e4i)T + 1.48e8iT^{2} \)
29 \( 1 - 2.74e4iT - 5.94e8T^{2} \)
31 \( 1 - 1.50e4T + 8.87e8T^{2} \)
37 \( 1 + (-2.15e4 + 2.15e4i)T - 2.56e9iT^{2} \)
41 \( 1 + 4.56e4T + 4.75e9T^{2} \)
43 \( 1 + (7.81e4 + 7.81e4i)T + 6.32e9iT^{2} \)
47 \( 1 + (1.99e3 - 1.99e3i)T - 1.07e10iT^{2} \)
53 \( 1 + (-1.06e5 - 1.06e5i)T + 2.21e10iT^{2} \)
59 \( 1 + 6.44e4iT - 4.21e10T^{2} \)
61 \( 1 + 3.20e5T + 5.15e10T^{2} \)
67 \( 1 + (3.50e4 - 3.50e4i)T - 9.04e10iT^{2} \)
71 \( 1 + 2.51e5T + 1.28e11T^{2} \)
73 \( 1 + (-3.26e5 - 3.26e5i)T + 1.51e11iT^{2} \)
79 \( 1 - 2.57e4iT - 2.43e11T^{2} \)
83 \( 1 + (1.23e5 + 1.23e5i)T + 3.26e11iT^{2} \)
89 \( 1 + 1.98e5iT - 4.96e11T^{2} \)
97 \( 1 + (7.38e5 - 7.38e5i)T - 8.32e11iT^{2} \)
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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

−20.01812020634312393210501414854, −18.60529288134754529078304040883, −16.22570877029162685586878433785, −15.27676046670092996757039886375, −13.60095063780865175225704173567, −12.33380259299754264795576076249, −9.916950986622585994428998207937, −8.952716843747104855976091979215, −5.31412258793927776665734698090, −3.01792865844079251944410104589, 3.03301789891226207024159411636, 6.55682589336542425352029894458, 7.82748322185501701614364511767, 10.28766199879950969075487559914, 12.99685235440476613277426172237, 13.66834065655658985892842290580, 14.98562356302003055064526946978, 16.78966728309287770840804268485, 18.47652757659238380147687681677, 19.52138599362696228196261969977

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