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Properties

Label 2-10-5.2-c6-0-1
Degree $2$
Conductor $10$
Sign $-0.245 - 0.969i$
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 + (−32.8 + 32.8i)3-s + 32i·4-s + (110. − 57.6i)5-s − 263.·6-s + (261. + 261. i)7-s + (−128 + 128i)8-s − 1.43e3i·9-s + (674. + 212. i)10-s + 155.·11-s + (−1.05e3 − 1.05e3i)12-s + (141. − 141. i)13-s + 2.09e3i·14-s + (−1.75e3 + 5.54e3i)15-s − 1.02e3·16-s + (1.78e3 + 1.78e3i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (−1.21 + 1.21i)3-s + 0.5i·4-s + (0.887 − 0.461i)5-s − 1.21·6-s + (0.763 + 0.763i)7-s + (−0.250 + 0.250i)8-s − 1.96i·9-s + (0.674 + 0.212i)10-s + 0.116·11-s + (−0.609 − 0.609i)12-s + (0.0642 − 0.0642i)13-s + 0.763i·14-s + (−0.518 + 1.64i)15-s − 0.250·16-s + (0.363 + 0.363i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.245 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.245 - 0.969i$
Analytic conductor: \(2.30054\)
Root analytic conductor: \(1.51675\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :3),\ -0.245 - 0.969i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.818487 + 1.05138i\)
\(L(\frac12)\) \(\approx\) \(0.818487 + 1.05138i\)
\(L(4)\) 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 \( 1 + (-110. + 57.6i)T \)
good3 \( 1 + (32.8 - 32.8i)T - 729iT^{2} \)
7 \( 1 + (-261. - 261. i)T + 1.17e5iT^{2} \)
11 \( 1 - 155.T + 1.77e6T^{2} \)
13 \( 1 + (-141. + 141. i)T - 4.82e6iT^{2} \)
17 \( 1 + (-1.78e3 - 1.78e3i)T + 2.41e7iT^{2} \)
19 \( 1 + 4.68e3iT - 4.70e7T^{2} \)
23 \( 1 + (-6.53e3 + 6.53e3i)T - 1.48e8iT^{2} \)
29 \( 1 + 1.14e3iT - 5.94e8T^{2} \)
31 \( 1 + 2.45e4T + 8.87e8T^{2} \)
37 \( 1 + (5.54e4 + 5.54e4i)T + 2.56e9iT^{2} \)
41 \( 1 - 1.00e5T + 4.75e9T^{2} \)
43 \( 1 + (6.86e4 - 6.86e4i)T - 6.32e9iT^{2} \)
47 \( 1 + (-9.97e4 - 9.97e4i)T + 1.07e10iT^{2} \)
53 \( 1 + (1.67e5 - 1.67e5i)T - 2.21e10iT^{2} \)
59 \( 1 + 2.29e5iT - 4.21e10T^{2} \)
61 \( 1 - 1.99e5T + 5.15e10T^{2} \)
67 \( 1 + (3.51e5 + 3.51e5i)T + 9.04e10iT^{2} \)
71 \( 1 + 2.71e5T + 1.28e11T^{2} \)
73 \( 1 + (-1.34e5 + 1.34e5i)T - 1.51e11iT^{2} \)
79 \( 1 + 3.32e5iT - 2.43e11T^{2} \)
83 \( 1 + (4.95e5 - 4.95e5i)T - 3.26e11iT^{2} \)
89 \( 1 + 1.93e4iT - 4.96e11T^{2} \)
97 \( 1 + (2.29e5 + 2.29e5i)T + 8.32e11iT^{2} \)
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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

−20.87995804690525154166284888370, −17.90936020953812380133273968836, −17.03880250524559356098605958828, −15.87332441704130923676646900702, −14.57825157421047814976578471701, −12.46936479330100885058268925480, −10.94691574864752900000782915430, −9.147040206795615514591754420888, −5.90067106305982312140505094735, −4.81669194207656479349920822367, 1.46282831215707304979656645823, 5.50091325445185230341178721994, 7.09315086984818395190683839637, 10.50947711456331724478886800176, 11.67987982685269922810073649935, 13.16239860269543271112097766507, 14.23731975018222010861677428032, 16.93141826483275154910021878681, 17.90463480815530726207714285177, 18.94982615469883833058516819265

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