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Properties

Label 2-10-5.3-c16-0-5
Degree $2$
Conductor $10$
Sign $-0.815 + 0.579i$
Analytic cond. $16.2324$
Root an. cond. $4.02895$
Motivic weight $16$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−128 + 128i)2-s + (−7.76e3 − 7.76e3i)3-s − 3.27e4i·4-s + (1.47e5 + 3.61e5i)5-s + 1.98e6·6-s + (7.60e6 − 7.60e6i)7-s + (4.19e6 + 4.19e6i)8-s + 7.74e7i·9-s + (−6.51e7 − 2.75e7i)10-s + 4.04e7·11-s + (−2.54e8 + 2.54e8i)12-s + (−6.20e8 − 6.20e8i)13-s + 1.94e9i·14-s + (1.66e9 − 3.94e9i)15-s − 1.07e9·16-s + (9.38e7 − 9.38e7i)17-s + ⋯
L(s)  = 1  + (−0.5 + 0.5i)2-s + (−1.18 − 1.18i)3-s − 0.5i·4-s + (0.376 + 0.926i)5-s + 1.18·6-s + (1.31 − 1.31i)7-s + (0.250 + 0.250i)8-s + 1.79i·9-s + (−0.651 − 0.275i)10-s + 0.188·11-s + (−0.591 + 0.591i)12-s + (−0.760 − 0.760i)13-s + 1.31i·14-s + (0.650 − 1.54i)15-s − 0.250·16-s + (0.0134 − 0.0134i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(17-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+8) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10\)    =    \(2 \cdot 5\)
Sign: $-0.815 + 0.579i$
Analytic conductor: \(16.2324\)
Root analytic conductor: \(4.02895\)
Motivic weight: \(16\)
Rational: no
Arithmetic: yes
Character: $\chi_{10} (3, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 10,\ (\ :8),\ -0.815 + 0.579i)\)

Particular Values

\(L(\frac{17}{2})\) \(\approx\) \(0.185095 - 0.580117i\)
\(L(\frac12)\) \(\approx\) \(0.185095 - 0.580117i\)
\(L(9)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (128 - 128i)T \)
5 \( 1 + (-1.47e5 - 3.61e5i)T \)
good3 \( 1 + (7.76e3 + 7.76e3i)T + 4.30e7iT^{2} \)
7 \( 1 + (-7.60e6 + 7.60e6i)T - 3.32e13iT^{2} \)
11 \( 1 - 4.04e7T + 4.59e16T^{2} \)
13 \( 1 + (6.20e8 + 6.20e8i)T + 6.65e17iT^{2} \)
17 \( 1 + (-9.38e7 + 9.38e7i)T - 4.86e19iT^{2} \)
19 \( 1 + 1.65e10iT - 2.88e20T^{2} \)
23 \( 1 + (-1.43e10 - 1.43e10i)T + 6.13e21iT^{2} \)
29 \( 1 - 6.31e10iT - 2.50e23T^{2} \)
31 \( 1 + 1.18e12T + 7.27e23T^{2} \)
37 \( 1 + (-2.24e12 + 2.24e12i)T - 1.23e25iT^{2} \)
41 \( 1 + 1.58e13T + 6.37e25T^{2} \)
43 \( 1 + (4.02e12 + 4.02e12i)T + 1.36e26iT^{2} \)
47 \( 1 + (1.43e12 - 1.43e12i)T - 5.66e26iT^{2} \)
53 \( 1 + (-1.53e13 - 1.53e13i)T + 3.87e27iT^{2} \)
59 \( 1 + 1.04e14iT - 2.15e28T^{2} \)
61 \( 1 - 2.05e13T + 3.67e28T^{2} \)
67 \( 1 + (-1.79e14 + 1.79e14i)T - 1.64e29iT^{2} \)
71 \( 1 + 8.87e14T + 4.16e29T^{2} \)
73 \( 1 + (3.41e14 + 3.41e14i)T + 6.50e29iT^{2} \)
79 \( 1 - 5.43e14iT - 2.30e30T^{2} \)
83 \( 1 + (1.98e15 + 1.98e15i)T + 5.07e30iT^{2} \)
89 \( 1 - 2.02e15iT - 1.54e31T^{2} \)
97 \( 1 + (3.78e15 - 3.78e15i)T - 6.14e31iT^{2} \)
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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

−17.01674575247074111942732955055, −14.69777017534290435855060122025, −13.41089058273761598894150926690, −11.39925155150511204802506735641, −10.50530437841590246115715542614, −7.60676003890777024474908724754, −6.90135234358541287895745569599, −5.24305236095037133899356050027, −1.67866513111496229610955226798, −0.33418997876644255069945674140, 1.68664110248150064709349436687, 4.53683573714926280731186391389, 5.55731199295705145750690911427, 8.660789997402745269115678243078, 9.844098074881659470954476156851, 11.46240819794389552605631660811, 12.18860392077433081122524098844, 14.87545556962474770998935491608, 16.40995794193407159635864441542, 17.24696344816275788666807663481

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