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Properties

Label 2-10-5.2-c10-0-3
Degree $2$
Conductor $10$
Sign $0.557 - 0.830i$
Analytic cond. $6.35357$
Root an. cond. $2.52062$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (16 + 16i)2-s + (57 − 57i)3-s + 512i·4-s + (2.92e3 + 1.10e3i)5-s + 1.82e3·6-s + (6.95e3 + 6.95e3i)7-s + (−8.19e3 + 8.19e3i)8-s + 5.25e4i·9-s + (2.92e4 + 6.44e4i)10-s + 7.52e4·11-s + (2.91e4 + 2.91e4i)12-s + (1.09e5 − 1.09e5i)13-s + 2.22e5i·14-s + (2.29e5 − 1.04e5i)15-s − 2.62e5·16-s + (−1.52e6 − 1.52e6i)17-s + ⋯
L(s)  = 1  + (0.5 + 0.5i)2-s + (0.234 − 0.234i)3-s + 0.5i·4-s + (0.936 + 0.352i)5-s + 0.234·6-s + (0.413 + 0.413i)7-s + (−0.250 + 0.250i)8-s + 0.889i·9-s + (0.292 + 0.644i)10-s + 0.467·11-s + (0.117 + 0.117i)12-s + (0.295 − 0.295i)13-s + 0.413i·14-s + (0.302 − 0.136i)15-s − 0.250·16-s + (−1.07 − 1.07i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.557 - 0.830i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.557 - 0.830i$
Analytic conductor: \(6.35357\)
Root analytic conductor: \(2.52062\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :5),\ 0.557 - 0.830i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.22479 + 1.18563i\)
\(L(\frac12)\) \(\approx\) \(2.22479 + 1.18563i\)
\(L(6)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-16 - 16i)T \)
5 \( 1 + (-2.92e3 - 1.10e3i)T \)
good3 \( 1 + (-57 + 57i)T - 5.90e4iT^{2} \)
7 \( 1 + (-6.95e3 - 6.95e3i)T + 2.82e8iT^{2} \)
11 \( 1 - 7.52e4T + 2.59e10T^{2} \)
13 \( 1 + (-1.09e5 + 1.09e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (1.52e6 + 1.52e6i)T + 2.01e12iT^{2} \)
19 \( 1 + 4.03e6iT - 6.13e12T^{2} \)
23 \( 1 + (7.12e5 - 7.12e5i)T - 4.14e13iT^{2} \)
29 \( 1 - 4.46e5iT - 4.20e14T^{2} \)
31 \( 1 + 2.90e7T + 8.19e14T^{2} \)
37 \( 1 + (9.11e5 + 9.11e5i)T + 4.80e15iT^{2} \)
41 \( 1 + 1.63e8T + 1.34e16T^{2} \)
43 \( 1 + (-1.18e8 + 1.18e8i)T - 2.16e16iT^{2} \)
47 \( 1 + (-2.76e8 - 2.76e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (-3.08e8 + 3.08e8i)T - 1.74e17iT^{2} \)
59 \( 1 + 9.40e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.35e9T + 7.13e17T^{2} \)
67 \( 1 + (-8.53e8 - 8.53e8i)T + 1.82e18iT^{2} \)
71 \( 1 - 2.82e9T + 3.25e18T^{2} \)
73 \( 1 + (2.75e9 - 2.75e9i)T - 4.29e18iT^{2} \)
79 \( 1 - 3.32e9iT - 9.46e18T^{2} \)
83 \( 1 + (-1.34e9 + 1.34e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 2.66e9iT - 3.11e19T^{2} \)
97 \( 1 + (5.26e8 + 5.26e8i)T + 7.37e19iT^{2} \)
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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

−18.39634524542139402580114584633, −17.26944608037313628008584437461, −15.60467157306474261588579157493, −14.11076234856294098963190874907, −13.21320892687923712486291325465, −11.14695353300958144043870223346, −8.948689522200678052404287025130, −6.98215040703774643879951698499, −5.16464369333576426631803464446, −2.37842261372478713179066510225, 1.58099731909072381970786310110, 4.02374903709250392214806328264, 6.13260724991758774708242829790, 8.987284424356253584939278611471, 10.48187388561386472360744690374, 12.32121513427175594119994418304, 13.78833395774184528742258799633, 14.91836961386299830575491805189, 16.89361649096381474158077676476, 18.24201118133087810164147784822

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