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Properties

Label 2-10-5.2-c12-0-5
Degree $2$
Conductor $10$
Sign $-0.887 - 0.460i$
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 + (59.4 − 59.4i)3-s + 2.04e3i·4-s + (3.81e3 − 1.51e4i)5-s − 3.80e3·6-s + (−5.93e4 − 5.93e4i)7-s + (6.55e4 − 6.55e4i)8-s + 5.24e5i·9-s + (−6.07e5 + 3.62e5i)10-s − 2.34e6·11-s + (1.21e5 + 1.21e5i)12-s + (−4.56e6 + 4.56e6i)13-s + 3.79e6i·14-s + (−6.73e5 − 1.12e6i)15-s − 4.19e6·16-s + (−1.56e7 − 1.56e7i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (0.0815 − 0.0815i)3-s + 0.5i·4-s + (0.244 − 0.969i)5-s − 0.0815·6-s + (−0.504 − 0.504i)7-s + (0.250 − 0.250i)8-s + 0.986i·9-s + (−0.607 + 0.362i)10-s − 1.32·11-s + (0.0407 + 0.0407i)12-s + (−0.946 + 0.946i)13-s + 0.504i·14-s + (−0.0591 − 0.0989i)15-s − 0.250·16-s + (−0.648 − 0.648i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (-0.887 - 0.460i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.887 - 0.460i$
Analytic conductor: \(9.13993\)
Root analytic conductor: \(3.02323\)
Motivic weight: \(12\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :6),\ -0.887 - 0.460i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.0591792 + 0.242500i\)
\(L(\frac12)\) \(\approx\) \(0.0591792 + 0.242500i\)
\(L(7)\) 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 \( 1 + (-3.81e3 + 1.51e4i)T \)
good3 \( 1 + (-59.4 + 59.4i)T - 5.31e5iT^{2} \)
7 \( 1 + (5.93e4 + 5.93e4i)T + 1.38e10iT^{2} \)
11 \( 1 + 2.34e6T + 3.13e12T^{2} \)
13 \( 1 + (4.56e6 - 4.56e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (1.56e7 + 1.56e7i)T + 5.82e14iT^{2} \)
19 \( 1 - 1.45e7iT - 2.21e15T^{2} \)
23 \( 1 + (-1.10e8 + 1.10e8i)T - 2.19e16iT^{2} \)
29 \( 1 + 5.34e7iT - 3.53e17T^{2} \)
31 \( 1 + 1.35e9T + 7.87e17T^{2} \)
37 \( 1 + (-7.96e8 - 7.96e8i)T + 6.58e18iT^{2} \)
41 \( 1 - 5.38e8T + 2.25e19T^{2} \)
43 \( 1 + (-7.82e9 + 7.82e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (-5.73e9 - 5.73e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (1.01e10 - 1.01e10i)T - 4.91e20iT^{2} \)
59 \( 1 + 5.72e10iT - 1.77e21T^{2} \)
61 \( 1 + 9.47e10T + 2.65e21T^{2} \)
67 \( 1 + (4.17e10 + 4.17e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 3.51e10T + 1.64e22T^{2} \)
73 \( 1 + (-1.08e11 + 1.08e11i)T - 2.29e22iT^{2} \)
79 \( 1 + 1.31e11iT - 5.90e22T^{2} \)
83 \( 1 + (-1.61e11 + 1.61e11i)T - 1.06e23iT^{2} \)
89 \( 1 - 8.98e11iT - 2.46e23T^{2} \)
97 \( 1 + (3.98e11 + 3.98e11i)T + 6.93e23iT^{2} \)
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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

−16.98055192107226411592662608991, −16.13213186903575147371356646350, −13.65444918201337984558430756853, −12.58576205685130306827938785854, −10.68546737398206318493348299126, −9.220021599710345925165727980599, −7.54775640114164268575407983309, −4.80958799438874033189395444415, −2.22782837048579175028284010082, −0.13131375008003543106880383191, 2.79890436474732723365171433338, 5.80164875050069037948858751200, 7.40041346121327137708973818524, 9.382552754113523290883916055773, 10.72197420585935931223165507256, 12.88462975801583261027924619256, 14.85488196735354521589004864230, 15.57359811802744842878817699220, 17.55843363865688368104234812490, 18.39270070602944680064089466510

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