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Properties

Label 2-10-5.2-c14-0-4
Degree $2$
Conductor $10$
Sign $0.420 + 0.907i$
Analytic cond. $12.4328$
Root an. cond. $3.52603$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−64 − 64i)2-s + (1.29e3 − 1.29e3i)3-s + 8.19e3i·4-s + (6.14e4 − 4.82e4i)5-s − 1.65e5·6-s + (9.05e5 + 9.05e5i)7-s + (5.24e5 − 5.24e5i)8-s + 1.42e6i·9-s + (−7.02e6 − 8.40e5i)10-s − 4.06e6·11-s + (1.06e7 + 1.06e7i)12-s + (8.29e7 − 8.29e7i)13-s − 1.15e8i·14-s + (1.70e7 − 1.42e8i)15-s − 6.71e7·16-s + (−2.30e7 − 2.30e7i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.592 − 0.592i)3-s + 0.5i·4-s + (0.786 − 0.618i)5-s − 0.592·6-s + (1.09 + 1.09i)7-s + (0.250 − 0.250i)8-s + 0.297i·9-s + (−0.702 − 0.0840i)10-s − 0.208·11-s + (0.296 + 0.296i)12-s + (1.32 − 1.32i)13-s − 1.09i·14-s + (0.0995 − 0.832i)15-s − 0.250·16-s + (−0.0562 − 0.0562i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (0.420 + 0.907i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.420 + 0.907i$
Analytic conductor: \(12.4328\)
Root analytic conductor: \(3.52603\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :7),\ 0.420 + 0.907i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(1.82237 - 1.16339i\)
\(L(\frac12)\) \(\approx\) \(1.82237 - 1.16339i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (64 + 64i)T \)
5 \( 1 + (-6.14e4 + 4.82e4i)T \)
good3 \( 1 + (-1.29e3 + 1.29e3i)T - 4.78e6iT^{2} \)
7 \( 1 + (-9.05e5 - 9.05e5i)T + 6.78e11iT^{2} \)
11 \( 1 + 4.06e6T + 3.79e14T^{2} \)
13 \( 1 + (-8.29e7 + 8.29e7i)T - 3.93e15iT^{2} \)
17 \( 1 + (2.30e7 + 2.30e7i)T + 1.68e17iT^{2} \)
19 \( 1 + 1.25e9iT - 7.99e17T^{2} \)
23 \( 1 + (2.88e9 - 2.88e9i)T - 1.15e19iT^{2} \)
29 \( 1 - 9.91e9iT - 2.97e20T^{2} \)
31 \( 1 + 2.55e9T + 7.56e20T^{2} \)
37 \( 1 + (4.48e10 + 4.48e10i)T + 9.01e21iT^{2} \)
41 \( 1 - 1.89e11T + 3.79e22T^{2} \)
43 \( 1 + (-1.27e11 + 1.27e11i)T - 7.38e22iT^{2} \)
47 \( 1 + (2.61e11 + 2.61e11i)T + 2.56e23iT^{2} \)
53 \( 1 + (1.34e12 - 1.34e12i)T - 1.37e24iT^{2} \)
59 \( 1 - 1.28e12iT - 6.19e24T^{2} \)
61 \( 1 - 6.26e11T + 9.87e24T^{2} \)
67 \( 1 + (-2.44e12 - 2.44e12i)T + 3.67e25iT^{2} \)
71 \( 1 - 3.25e12T + 8.27e25T^{2} \)
73 \( 1 + (-8.34e12 + 8.34e12i)T - 1.22e26iT^{2} \)
79 \( 1 - 7.48e12iT - 3.68e26T^{2} \)
83 \( 1 + (3.14e13 - 3.14e13i)T - 7.36e26iT^{2} \)
89 \( 1 - 1.25e13iT - 1.95e27T^{2} \)
97 \( 1 + (1.05e14 + 1.05e14i)T + 6.52e27iT^{2} \)
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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

−17.70672237752863698141140568385, −15.68550234615432491639347486102, −13.78864644673408337182307806073, −12.69163437059622251827040336048, −10.96063146202238589817635896394, −8.949247720358291296531974498423, −8.046203289802761964478675780374, −5.39597994778234086553115408257, −2.49432145868979562775025812050, −1.31002183821587873281820054010, 1.57356108671588668059683306035, 4.06651761486263025638619788656, 6.39519610993865067928942176765, 8.215798201808670735207438097380, 9.781385785861994400821150058549, 10.99035445707375935390740603523, 13.99442829541361102680983146845, 14.49099664954235366357090276211, 16.22239433339979920516885351480, 17.58918933375128985500612975275

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