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Properties

Label 2-17-17.13-c15-0-20
Degree $2$
Conductor $17$
Sign $-0.573 - 0.819i$
Analytic cond. $24.2578$
Root an. cond. $4.92523$
Motivic weight $15$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 274. i·2-s + (931. + 931. i)3-s − 4.26e4·4-s + (1.29e5 + 1.29e5i)5-s + (2.55e5 − 2.55e5i)6-s + (1.51e5 − 1.51e5i)7-s + 2.72e6i·8-s − 1.26e7i·9-s + (3.55e7 − 3.55e7i)10-s + (−5.86e7 + 5.86e7i)11-s + (−3.97e7 − 3.97e7i)12-s − 4.27e8·13-s + (−4.16e7 − 4.16e7i)14-s + 2.41e8i·15-s − 6.51e8·16-s + (4.95e8 − 1.61e9i)17-s + ⋯
L(s)  = 1  − 1.51i·2-s + (0.245 + 0.245i)3-s − 1.30·4-s + (0.740 + 0.740i)5-s + (0.373 − 0.373i)6-s + (0.0696 − 0.0696i)7-s + 0.458i·8-s − 0.879i·9-s + (1.12 − 1.12i)10-s + (−0.907 + 0.907i)11-s + (−0.320 − 0.320i)12-s − 1.88·13-s + (−0.105 − 0.105i)14-s + 0.364i·15-s − 0.606·16-s + (0.292 − 0.956i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & (-0.573 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.573 - 0.819i$
Analytic conductor: \(24.2578\)
Root analytic conductor: \(4.92523\)
Motivic weight: \(15\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :15/2),\ -0.573 - 0.819i)\)

Particular Values

\(L(8)\) \(\approx\) \(0.6372882203\)
\(L(\frac12)\) \(\approx\) \(0.6372882203\)
\(L(\frac{17}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-4.95e8 + 1.61e9i)T \)
good2 \( 1 + 274. iT - 3.27e4T^{2} \)
3 \( 1 + (-931. - 931. i)T + 1.43e7iT^{2} \)
5 \( 1 + (-1.29e5 - 1.29e5i)T + 3.05e10iT^{2} \)
7 \( 1 + (-1.51e5 + 1.51e5i)T - 4.74e12iT^{2} \)
11 \( 1 + (5.86e7 - 5.86e7i)T - 4.17e15iT^{2} \)
13 \( 1 + 4.27e8T + 5.11e16T^{2} \)
19 \( 1 + 4.87e9iT - 1.51e19T^{2} \)
23 \( 1 + (1.56e10 - 1.56e10i)T - 2.66e20iT^{2} \)
29 \( 1 + (-2.87e10 - 2.87e10i)T + 8.62e21iT^{2} \)
31 \( 1 + (9.32e10 + 9.32e10i)T + 2.34e22iT^{2} \)
37 \( 1 + (-2.66e11 - 2.66e11i)T + 3.33e23iT^{2} \)
41 \( 1 + (9.16e11 - 9.16e11i)T - 1.55e24iT^{2} \)
43 \( 1 + 1.52e12iT - 3.17e24T^{2} \)
47 \( 1 + 2.80e12T + 1.20e25T^{2} \)
53 \( 1 + 5.81e12iT - 7.31e25T^{2} \)
59 \( 1 + 1.31e13iT - 3.65e26T^{2} \)
61 \( 1 + (-1.54e13 + 1.54e13i)T - 6.02e26iT^{2} \)
67 \( 1 + 7.02e13T + 2.46e27T^{2} \)
71 \( 1 + (-6.24e13 - 6.24e13i)T + 5.87e27iT^{2} \)
73 \( 1 + (-5.34e13 - 5.34e13i)T + 8.90e27iT^{2} \)
79 \( 1 + (-6.74e12 + 6.74e12i)T - 2.91e28iT^{2} \)
83 \( 1 + 9.52e13iT - 6.11e28T^{2} \)
89 \( 1 + 5.02e14T + 1.74e29T^{2} \)
97 \( 1 + (8.48e14 + 8.48e14i)T + 6.33e29iT^{2} \)
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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

−14.10053577828751684302048729705, −12.71845994981888480277691817247, −11.57205794854390289754934464366, −9.939190696845787353210881342530, −9.681681787322901172019818394663, −7.10551680146720919349736527107, −4.79829075641712556444081847759, −2.97546957605328768274163681194, −2.16209910263528094284926927613, −0.18694843718722410720288721447, 2.12338846045814156605395347502, 4.95041007374619239980047995935, 5.85468422551909847403241019669, 7.64621447363170041976005750124, 8.488309619216986284360155967875, 10.18214054908619769256879332027, 12.61931377498349549190682650662, 13.81834897930555872811629567851, 14.75166711349495426852428743410, 16.41537275471463556522569033377

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