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Properties

Label 2-10-5.3-c14-0-6
Degree $2$
Conductor $10$
Sign $-0.0500 - 0.998i$
Analytic cond. $12.4328$
Root an. cond. $3.52603$
Motivic weight $14$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (64 − 64i)2-s + (−2.73e3 − 2.73e3i)3-s − 8.19e3i·4-s + (−7.68e4 − 1.41e4i)5-s − 3.50e5·6-s + (3.90e5 − 3.90e5i)7-s + (−5.24e5 − 5.24e5i)8-s + 1.02e7i·9-s + (−5.82e6 + 4.01e6i)10-s + 2.58e7·11-s + (−2.24e7 + 2.24e7i)12-s + (−4.31e7 − 4.31e7i)13-s − 5.00e7i·14-s + (1.71e8 + 2.49e8i)15-s − 6.71e7·16-s + (−2.86e8 + 2.86e8i)17-s + ⋯
L(s)  = 1  + (0.5 − 0.5i)2-s + (−1.25 − 1.25i)3-s − 0.5i·4-s + (−0.983 − 0.180i)5-s − 1.25·6-s + (0.474 − 0.474i)7-s + (−0.250 − 0.250i)8-s + 2.13i·9-s + (−0.582 + 0.401i)10-s + 1.32·11-s + (−0.626 + 0.626i)12-s + (−0.688 − 0.688i)13-s − 0.474i·14-s + (1.00 + 1.45i)15-s − 0.250·16-s + (−0.698 + 0.698i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0500 - 0.998i)\, \overline{\Lambda}(15-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+7) \, L(s)\cr =\mathstrut & (-0.0500 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.0500 - 0.998i$
Analytic conductor: \(12.4328\)
Root analytic conductor: \(3.52603\)
Motivic weight: \(14\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :7),\ -0.0500 - 0.998i)\)

Particular Values

\(L(\frac{15}{2})\) \(\approx\) \(0.251311 + 0.264233i\)
\(L(\frac12)\) \(\approx\) \(0.251311 + 0.264233i\)
\(L(8)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-64 + 64i)T \)
5 \( 1 + (7.68e4 + 1.41e4i)T \)
good3 \( 1 + (2.73e3 + 2.73e3i)T + 4.78e6iT^{2} \)
7 \( 1 + (-3.90e5 + 3.90e5i)T - 6.78e11iT^{2} \)
11 \( 1 - 2.58e7T + 3.79e14T^{2} \)
13 \( 1 + (4.31e7 + 4.31e7i)T + 3.93e15iT^{2} \)
17 \( 1 + (2.86e8 - 2.86e8i)T - 1.68e17iT^{2} \)
19 \( 1 - 9.14e8iT - 7.99e17T^{2} \)
23 \( 1 + (3.49e9 + 3.49e9i)T + 1.15e19iT^{2} \)
29 \( 1 + 9.99e9iT - 2.97e20T^{2} \)
31 \( 1 + 2.36e10T + 7.56e20T^{2} \)
37 \( 1 + (-2.78e10 + 2.78e10i)T - 9.01e21iT^{2} \)
41 \( 1 - 4.76e10T + 3.79e22T^{2} \)
43 \( 1 + (-6.06e10 - 6.06e10i)T + 7.38e22iT^{2} \)
47 \( 1 + (6.16e11 - 6.16e11i)T - 2.56e23iT^{2} \)
53 \( 1 + (3.69e11 + 3.69e11i)T + 1.37e24iT^{2} \)
59 \( 1 - 1.87e12iT - 6.19e24T^{2} \)
61 \( 1 + 3.25e12T + 9.87e24T^{2} \)
67 \( 1 + (-5.83e12 + 5.83e12i)T - 3.67e25iT^{2} \)
71 \( 1 + 1.14e13T + 8.27e25T^{2} \)
73 \( 1 + (3.16e12 + 3.16e12i)T + 1.22e26iT^{2} \)
79 \( 1 + 2.36e13iT - 3.68e26T^{2} \)
83 \( 1 + (-1.83e13 - 1.83e13i)T + 7.36e26iT^{2} \)
89 \( 1 + 6.00e13iT - 1.95e27T^{2} \)
97 \( 1 + (-1.30e12 + 1.30e12i)T - 6.52e27iT^{2} \)
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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

−16.59012033672099038690145637900, −14.49414008468513870260031273033, −12.71726589173041950308263336945, −11.93348830965462574362000861870, −10.84735680937829777429809139479, −7.76365492837137365793906258856, −6.22766463821128906402962353909, −4.36062418482321437494894345784, −1.50925619926330794683922446094, −0.16444163185227974997542651619, 3.94454732092718957531468894578, 5.01677580550318745526674216927, 6.78472227135461912061099268839, 9.218356322163854348230759709541, 11.37460052703134687580597166483, 11.87979541302265437759369074474, 14.62609218951979072230429137001, 15.63052836594942982001451554705, 16.61880598309284056877394383175, 17.81987259955554861576517665512

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