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Properties

Label 2-10-5.3-c10-0-0
Degree $2$
Conductor $10$
Sign $0.357 - 0.933i$
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 + (−266. − 266. i)3-s − 512i·4-s + (2.58e3 + 1.75e3i)5-s + 8.52e3·6-s + (−1.30e4 + 1.30e4i)7-s + (8.19e3 + 8.19e3i)8-s + 8.29e4i·9-s + (−6.94e4 + 1.32e4i)10-s + 2.39e5·11-s + (−1.36e5 + 1.36e5i)12-s + (2.87e5 + 2.87e5i)13-s − 4.16e5i·14-s + (−2.19e5 − 1.15e6i)15-s − 2.62e5·16-s + (−8.84e5 + 8.84e5i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−1.09 − 1.09i)3-s − 0.5i·4-s + (0.826 + 0.562i)5-s + 1.09·6-s + (−0.774 + 0.774i)7-s + (0.250 + 0.250i)8-s + 1.40i·9-s + (−0.694 + 0.132i)10-s + 1.48·11-s + (−0.548 + 0.548i)12-s + (0.775 + 0.775i)13-s − 0.774i·14-s + (−0.289 − 1.52i)15-s − 0.250·16-s + (−0.622 + 0.622i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.357 - 0.933i$
Analytic conductor: \(6.35357\)
Root analytic conductor: \(2.52062\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :5),\ 0.357 - 0.933i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.683020 + 0.469798i\)
\(L(\frac12)\) \(\approx\) \(0.683020 + 0.469798i\)
\(L(6)\) 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 \( 1 + (-2.58e3 - 1.75e3i)T \)
good3 \( 1 + (266. + 266. i)T + 5.90e4iT^{2} \)
7 \( 1 + (1.30e4 - 1.30e4i)T - 2.82e8iT^{2} \)
11 \( 1 - 2.39e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.87e5 - 2.87e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (8.84e5 - 8.84e5i)T - 2.01e12iT^{2} \)
19 \( 1 - 1.97e5iT - 6.13e12T^{2} \)
23 \( 1 + (3.55e6 + 3.55e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 3.57e7iT - 4.20e14T^{2} \)
31 \( 1 - 9.65e6T + 8.19e14T^{2} \)
37 \( 1 + (-1.34e7 + 1.34e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 1.08e8T + 1.34e16T^{2} \)
43 \( 1 + (-8.62e7 - 8.62e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (1.76e8 - 1.76e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (1.48e8 + 1.48e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 7.98e8iT - 5.11e17T^{2} \)
61 \( 1 + 4.24e8T + 7.13e17T^{2} \)
67 \( 1 + (-1.12e9 + 1.12e9i)T - 1.82e18iT^{2} \)
71 \( 1 + 2.60e9T + 3.25e18T^{2} \)
73 \( 1 + (-8.51e8 - 8.51e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 4.56e8iT - 9.46e18T^{2} \)
83 \( 1 + (-7.60e8 - 7.60e8i)T + 1.55e19iT^{2} \)
89 \( 1 - 2.03e9iT - 3.11e19T^{2} \)
97 \( 1 + (-8.73e9 + 8.73e9i)T - 7.37e19iT^{2} \)
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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

−18.41230586910919190004928518970, −17.49786528806568927850532550481, −16.33231700181081605965444549256, −14.29250627122603055254240664016, −12.66565738975012803747731676038, −11.13850336522428829352787152563, −9.175852401431695852055325977474, −6.60473614768392842161960692277, −6.15528575952692494364544430999, −1.55854944776641502174177713553, 0.68891804545024653044520043589, 4.09718273821030828859048374502, 6.15629448225037116201783672916, 9.316681832365865023410814193161, 10.26515069507746744049367123442, 11.68115846619159780800242547911, 13.45474964321641245001613029429, 15.91663941162724097642452908340, 16.93613397803105967722348549772, 17.67089803775526154600628112301

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