Properties

Label 2-10-5.3-c12-0-2
Degree $2$
Conductor $10$
Sign $0.731 - 0.682i$
Analytic cond. $9.13993$
Root an. cond. $3.02323$
Motivic weight $12$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (32 − 32i)2-s + (693. + 693. i)3-s − 2.04e3i·4-s + (7.74e3 + 1.35e4i)5-s + 4.43e4·6-s + (−1.08e5 + 1.08e5i)7-s + (−6.55e4 − 6.55e4i)8-s + 4.29e5i·9-s + (6.82e5 + 1.86e5i)10-s + 2.46e6·11-s + (1.41e6 − 1.41e6i)12-s + (7.82e5 + 7.82e5i)13-s + 6.94e6i·14-s + (−4.03e6 + 1.47e7i)15-s − 4.19e6·16-s + (2.16e7 − 2.16e7i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (0.950 + 0.950i)3-s − 0.5i·4-s + (0.495 + 0.868i)5-s + 0.950·6-s + (−0.922 + 0.922i)7-s + (−0.250 − 0.250i)8-s + 0.808i·9-s + (0.682 + 0.186i)10-s + 1.39·11-s + (0.475 − 0.475i)12-s + (0.162 + 0.162i)13-s + 0.922i·14-s + (−0.354 + 1.29i)15-s − 0.250·16-s + (0.895 − 0.895i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.731 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.731 - 0.682i$
Analytic conductor: \(9.13993\)
Root analytic conductor: \(3.02323\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :6),\ 0.731 - 0.682i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(2.77699 + 1.09389i\)
\(L(\frac12)\) \(\approx\) \(2.77699 + 1.09389i\)
\(L(7)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-32 + 32i)T \)
5 \( 1 + (-7.74e3 - 1.35e4i)T \)
good3 \( 1 + (-693. - 693. i)T + 5.31e5iT^{2} \)
7 \( 1 + (1.08e5 - 1.08e5i)T - 1.38e10iT^{2} \)
11 \( 1 - 2.46e6T + 3.13e12T^{2} \)
13 \( 1 + (-7.82e5 - 7.82e5i)T + 2.32e13iT^{2} \)
17 \( 1 + (-2.16e7 + 2.16e7i)T - 5.82e14iT^{2} \)
19 \( 1 - 1.52e7iT - 2.21e15T^{2} \)
23 \( 1 + (2.05e8 + 2.05e8i)T + 2.19e16iT^{2} \)
29 \( 1 + 2.06e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.07e8T + 7.87e17T^{2} \)
37 \( 1 + (-1.52e8 + 1.52e8i)T - 6.58e18iT^{2} \)
41 \( 1 - 7.91e9T + 2.25e19T^{2} \)
43 \( 1 + (5.63e9 + 5.63e9i)T + 3.99e19iT^{2} \)
47 \( 1 + (-2.56e9 + 2.56e9i)T - 1.16e20iT^{2} \)
53 \( 1 + (1.46e10 + 1.46e10i)T + 4.91e20iT^{2} \)
59 \( 1 - 5.45e10iT - 1.77e21T^{2} \)
61 \( 1 + 1.89e9T + 2.65e21T^{2} \)
67 \( 1 + (5.73e10 - 5.73e10i)T - 8.18e21iT^{2} \)
71 \( 1 - 1.06e11T + 1.64e22T^{2} \)
73 \( 1 + (5.40e10 + 5.40e10i)T + 2.29e22iT^{2} \)
79 \( 1 + 7.36e10iT - 5.90e22T^{2} \)
83 \( 1 + (-3.76e11 - 3.76e11i)T + 1.06e23iT^{2} \)
89 \( 1 + 4.32e11iT - 2.46e23T^{2} \)
97 \( 1 + (-9.08e11 + 9.08e11i)T - 6.93e23iT^{2} \)
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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

−18.51098138354754953270731656272, −16.15853099005251925033899001006, −14.77557639524702213691685940648, −14.04426344465367613119279231391, −12.03863068174093331963221000087, −10.07082656710922486120536858584, −9.177446929154321283771888924477, −6.20105399245093140862734670127, −3.75844889258990611288352339447, −2.54224139153016316943385007744, 1.36819003458998788403911906946, 3.70298978436624568070546671769, 6.29043934589467283317071053813, 7.83950051754511779914798100506, 9.430332261412995799504297113276, 12.43634211877720896198455027624, 13.43957609558400373527157010389, 14.30604006430513612356641993041, 16.33293176696235782132801983947, 17.46585303315520556329340680276

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