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Properties

Label 2-5-5.4-c17-0-5
Degree $2$
Conductor $5$
Sign $-0.396 + 0.917i$
Analytic cond. $9.16110$
Root an. cond. $3.02673$
Motivic weight $17$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 512. i·2-s + 1.82e4i·3-s − 1.31e5·4-s + (3.46e5 − 8.01e5i)5-s + 9.35e6·6-s − 1.34e7i·7-s + 3.36e5i·8-s − 2.03e8·9-s + (−4.11e8 − 1.77e8i)10-s + 8.20e8·11-s − 2.40e9i·12-s − 2.92e9i·13-s − 6.91e9·14-s + (1.46e10 + 6.32e9i)15-s − 1.70e10·16-s − 3.73e10i·17-s + ⋯
L(s)  = 1  − 1.41i·2-s + 1.60i·3-s − 1.00·4-s + (0.396 − 0.917i)5-s + 2.27·6-s − 0.883i·7-s + 0.00708i·8-s − 1.57·9-s + (−1.29 − 0.561i)10-s + 1.15·11-s − 1.61i·12-s − 0.995i·13-s − 1.25·14-s + (1.47 + 0.637i)15-s − 0.994·16-s − 1.30i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.396 + 0.917i)\, \overline{\Lambda}(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & (-0.396 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5\)
Sign: $-0.396 + 0.917i$
Analytic conductor: \(9.16110\)
Root analytic conductor: \(3.02673\)
Motivic weight: \(17\)
Rational: no
Arithmetic: yes
Character: $\chi_{5} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 5,\ (\ :17/2),\ -0.396 + 0.917i)\)

Particular Values

\(L(9)\) \(\approx\) \(0.934291 - 1.42184i\)
\(L(\frac12)\) \(\approx\) \(0.934291 - 1.42184i\)
\(L(\frac{19}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-3.46e5 + 8.01e5i)T \)
good2 \( 1 + 512. iT - 1.31e5T^{2} \)
3 \( 1 - 1.82e4iT - 1.29e8T^{2} \)
7 \( 1 + 1.34e7iT - 2.32e14T^{2} \)
11 \( 1 - 8.20e8T + 5.05e17T^{2} \)
13 \( 1 + 2.92e9iT - 8.65e18T^{2} \)
17 \( 1 + 3.73e10iT - 8.27e20T^{2} \)
19 \( 1 - 3.00e10T + 5.48e21T^{2} \)
23 \( 1 - 4.40e11iT - 1.41e23T^{2} \)
29 \( 1 - 1.80e12T + 7.25e24T^{2} \)
31 \( 1 - 2.90e12T + 2.25e25T^{2} \)
37 \( 1 - 3.31e12iT - 4.56e26T^{2} \)
41 \( 1 + 7.87e13T + 2.61e27T^{2} \)
43 \( 1 - 4.61e12iT - 5.87e27T^{2} \)
47 \( 1 - 1.39e14iT - 2.66e28T^{2} \)
53 \( 1 - 4.39e13iT - 2.05e29T^{2} \)
59 \( 1 + 3.18e14T + 1.27e30T^{2} \)
61 \( 1 - 7.17e14T + 2.24e30T^{2} \)
67 \( 1 + 2.87e15iT - 1.10e31T^{2} \)
71 \( 1 + 3.39e14T + 2.96e31T^{2} \)
73 \( 1 - 8.49e15iT - 4.74e31T^{2} \)
79 \( 1 - 1.05e16T + 1.81e32T^{2} \)
83 \( 1 + 2.86e16iT - 4.21e32T^{2} \)
89 \( 1 - 6.62e16T + 1.37e33T^{2} \)
97 \( 1 - 2.94e16iT - 5.95e33T^{2} \)
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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

−19.98659418950675523915850896150, −17.30497796478639977573718878478, −15.90828146563584397004803387383, −13.70823142126381458813132845778, −11.71392849450218946826649132779, −10.19603144294750652297749115150, −9.296866998457131771775053845251, −4.77557233054311840163398467213, −3.44869972202378375392138932058, −0.894398863380974756150534442666, 2.00376158438617532957011914310, 6.17580734080903143768899000099, 6.83204213793252780556606918871, 8.548526298404405942343326454875, 11.90792624302271829063243322375, 13.85123753016774665154034295070, 14.86236295140120010930588680825, 17.02959175418687169427174014080, 18.18968761962573107769955028632, 19.21668909675786481903118149209

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