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Properties

Label 2-10-5.2-c18-0-2
Degree $2$
Conductor $10$
Sign $-0.957 - 0.289i$
Analytic cond. $20.5386$
Root an. cond. $4.53195$
Motivic weight $18$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−256 − 256i)2-s + (−1.28e4 + 1.28e4i)3-s + 1.31e5i·4-s + (−1.20e5 + 1.94e6i)5-s + 6.56e6·6-s + (2.37e7 + 2.37e7i)7-s + (3.35e7 − 3.35e7i)8-s + 5.89e7i·9-s + (5.29e8 − 4.68e8i)10-s + 1.35e9·11-s + (−1.67e9 − 1.67e9i)12-s + (6.09e8 − 6.09e8i)13-s − 1.21e10i·14-s + (−2.34e10 − 2.65e10i)15-s − 1.71e10·16-s + (4.65e10 + 4.65e10i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−0.651 + 0.651i)3-s + 0.5i·4-s + (−0.0614 + 0.998i)5-s + 0.651·6-s + (0.588 + 0.588i)7-s + (0.250 − 0.250i)8-s + 0.152i·9-s + (0.529 − 0.468i)10-s + 0.575·11-s + (−0.325 − 0.325i)12-s + (0.0574 − 0.0574i)13-s − 0.588i·14-s + (−0.609 − 0.689i)15-s − 0.250·16-s + (0.392 + 0.392i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (-0.957 - 0.289i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.957 - 0.289i$
Analytic conductor: \(20.5386\)
Root analytic conductor: \(4.53195\)
Motivic weight: \(18\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :9),\ -0.957 - 0.289i)\)

Particular Values

\(L(\frac{19}{2})\) \(\approx\) \(0.116762 + 0.790399i\)
\(L(\frac12)\) \(\approx\) \(0.116762 + 0.790399i\)
\(L(10)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (256 + 256i)T \)
5 \( 1 + (1.20e5 - 1.94e6i)T \)
good3 \( 1 + (1.28e4 - 1.28e4i)T - 3.87e8iT^{2} \)
7 \( 1 + (-2.37e7 - 2.37e7i)T + 1.62e15iT^{2} \)
11 \( 1 - 1.35e9T + 5.55e18T^{2} \)
13 \( 1 + (-6.09e8 + 6.09e8i)T - 1.12e20iT^{2} \)
17 \( 1 + (-4.65e10 - 4.65e10i)T + 1.40e22iT^{2} \)
19 \( 1 - 3.62e11iT - 1.04e23T^{2} \)
23 \( 1 + (1.92e12 - 1.92e12i)T - 3.24e24iT^{2} \)
29 \( 1 + 4.68e12iT - 2.10e26T^{2} \)
31 \( 1 + 3.88e12T + 6.99e26T^{2} \)
37 \( 1 + (3.71e13 + 3.71e13i)T + 1.68e28iT^{2} \)
41 \( 1 - 5.30e14T + 1.07e29T^{2} \)
43 \( 1 + (-4.20e14 + 4.20e14i)T - 2.52e29iT^{2} \)
47 \( 1 + (1.51e15 + 1.51e15i)T + 1.25e30iT^{2} \)
53 \( 1 + (3.95e15 - 3.95e15i)T - 1.08e31iT^{2} \)
59 \( 1 + 5.16e15iT - 7.50e31T^{2} \)
61 \( 1 + 1.53e16T + 1.36e32T^{2} \)
67 \( 1 + (-1.32e16 - 1.32e16i)T + 7.40e32iT^{2} \)
71 \( 1 - 8.13e16T + 2.10e33T^{2} \)
73 \( 1 + (5.71e16 - 5.71e16i)T - 3.46e33iT^{2} \)
79 \( 1 + 3.79e16iT - 1.43e34T^{2} \)
83 \( 1 + (4.95e16 - 4.95e16i)T - 3.49e34iT^{2} \)
89 \( 1 - 3.67e17iT - 1.22e35T^{2} \)
97 \( 1 + (4.58e17 + 4.58e17i)T + 5.77e35iT^{2} \)
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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

−17.16079461620605082999113199115, −15.72306977207968729894693305183, −14.25615422379627796189202719180, −11.96524382593072527133485619086, −10.96572095447475792301548494052, −9.823468158126415750420343119005, −7.87673302105684876587203811979, −5.80851105904478373027136219657, −3.82265007604006460586526931310, −1.92829329229319152245576252073, 0.40026339416724365821679735511, 1.35297204937398453685822777757, 4.62631369918866695813951007140, 6.27309661025683775276048137365, 7.75513699199391474187604698921, 9.292231915309935343430330083391, 11.28413471962274787240667320430, 12.63300427474196733315788085715, 14.26748551761779314545462171755, 16.11998563262809800866481163544

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