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Properties

Label 2-10-5.2-c12-0-2
Degree $2$
Conductor $10$
Sign $0.125 + 0.992i$
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 + (−864. + 864. i)3-s + 2.04e3i·4-s + (−1.46e4 + 5.47e3i)5-s + 5.53e4·6-s + (4.33e4 + 4.33e4i)7-s + (6.55e4 − 6.55e4i)8-s − 9.64e5i·9-s + (6.43e5 + 2.93e5i)10-s − 2.60e6·11-s + (−1.77e6 − 1.77e6i)12-s + (4.93e6 − 4.93e6i)13-s − 2.77e6i·14-s + (7.92e6 − 1.73e7i)15-s − 4.19e6·16-s + (2.23e7 + 2.23e7i)17-s + ⋯
L(s)  = 1  + (−0.5 − 0.5i)2-s + (−1.18 + 1.18i)3-s + 0.5i·4-s + (−0.936 + 0.350i)5-s + 1.18·6-s + (0.368 + 0.368i)7-s + (0.250 − 0.250i)8-s − 1.81i·9-s + (0.643 + 0.293i)10-s − 1.47·11-s + (−0.593 − 0.593i)12-s + (1.02 − 1.02i)13-s − 0.368i·14-s + (0.695 − 1.52i)15-s − 0.250·16-s + (0.927 + 0.927i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+6) \, L(s)\cr =\mathstrut & (0.125 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $0.125 + 0.992i$
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: \((2,\ 10,\ (\ :6),\ 0.125 + 0.992i)\)

Particular Values

\(L(\frac{13}{2})\) \(\approx\) \(0.195300 - 0.172106i\)
\(L(\frac12)\) \(\approx\) \(0.195300 - 0.172106i\)
\(L(7)\) 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 \( 1 + (1.46e4 - 5.47e3i)T \)
good3 \( 1 + (864. - 864. i)T - 5.31e5iT^{2} \)
7 \( 1 + (-4.33e4 - 4.33e4i)T + 1.38e10iT^{2} \)
11 \( 1 + 2.60e6T + 3.13e12T^{2} \)
13 \( 1 + (-4.93e6 + 4.93e6i)T - 2.32e13iT^{2} \)
17 \( 1 + (-2.23e7 - 2.23e7i)T + 5.82e14iT^{2} \)
19 \( 1 - 1.32e6iT - 2.21e15T^{2} \)
23 \( 1 + (2.79e7 - 2.79e7i)T - 2.19e16iT^{2} \)
29 \( 1 + 2.39e8iT - 3.53e17T^{2} \)
31 \( 1 + 1.35e8T + 7.87e17T^{2} \)
37 \( 1 + (3.51e9 + 3.51e9i)T + 6.58e18iT^{2} \)
41 \( 1 + 3.98e9T + 2.25e19T^{2} \)
43 \( 1 + (-1.65e9 + 1.65e9i)T - 3.99e19iT^{2} \)
47 \( 1 + (-5.37e9 - 5.37e9i)T + 1.16e20iT^{2} \)
53 \( 1 + (6.55e9 - 6.55e9i)T - 4.91e20iT^{2} \)
59 \( 1 + 6.32e10iT - 1.77e21T^{2} \)
61 \( 1 + 4.51e8T + 2.65e21T^{2} \)
67 \( 1 + (-6.54e10 - 6.54e10i)T + 8.18e21iT^{2} \)
71 \( 1 + 4.22e9T + 1.64e22T^{2} \)
73 \( 1 + (-1.56e11 + 1.56e11i)T - 2.29e22iT^{2} \)
79 \( 1 - 2.29e10iT - 5.90e22T^{2} \)
83 \( 1 + (6.14e10 - 6.14e10i)T - 1.06e23iT^{2} \)
89 \( 1 + 6.90e11iT - 2.46e23T^{2} \)
97 \( 1 + (-2.17e11 - 2.17e11i)T + 6.93e23iT^{2} \)
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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

−17.60258953454193009305208712419, −16.05698995971127084527898229648, −15.33771271480493986078266134942, −12.39994368763210670312966460848, −11.01514494513360680963244315163, −10.35690840603705616179139368274, −8.136168169743852477616230364542, −5.45460944213681970773809645796, −3.59577233178902612710526690823, −0.21080378252594763322600285028, 1.13720577922247383968848656870, 5.18957528172494143212878399873, 6.98961309608344212306579316423, 8.108320312697836686825001764462, 10.87087583886149471702976791765, 12.04243387235089118405943991354, 13.60554730989446000294537826008, 15.89705623925252035471152353761, 16.83056121596103342817408425754, 18.31636471477655896078479659348

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