Properties

Label 2-17-17.4-c9-0-7
Degree $2$
Conductor $17$
Sign $-0.687 + 0.726i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.8i·2-s + (−144. + 144. i)3-s + 346.·4-s + (−85.3 + 85.3i)5-s + (1.86e3 + 1.86e3i)6-s + (−3.20e3 − 3.20e3i)7-s − 1.10e4i·8-s − 2.22e4i·9-s + (1.09e3 + 1.09e3i)10-s + (−4.09e4 − 4.09e4i)11-s + (−5.01e4 + 5.01e4i)12-s − 5.07e4·13-s + (−4.11e4 + 4.11e4i)14-s − 2.47e4i·15-s + 3.55e4·16-s + (−5.15e4 − 3.40e5i)17-s + ⋯
L(s)  = 1  − 0.568i·2-s + (−1.03 + 1.03i)3-s + 0.677·4-s + (−0.0610 + 0.0610i)5-s + (0.586 + 0.586i)6-s + (−0.503 − 0.503i)7-s − 0.953i·8-s − 1.12i·9-s + (0.0346 + 0.0346i)10-s + (−0.842 − 0.842i)11-s + (−0.698 + 0.698i)12-s − 0.493·13-s + (−0.286 + 0.286i)14-s − 0.125i·15-s + 0.135·16-s + (−0.149 − 0.988i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.687 + 0.726i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.687 + 0.726i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.687 + 0.726i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.245298 - 0.569693i\)
\(L(\frac12)\) \(\approx\) \(0.245298 - 0.569693i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (5.15e4 + 3.40e5i)T \)
good2 \( 1 + 12.8iT - 512T^{2} \)
3 \( 1 + (144. - 144. i)T - 1.96e4iT^{2} \)
5 \( 1 + (85.3 - 85.3i)T - 1.95e6iT^{2} \)
7 \( 1 + (3.20e3 + 3.20e3i)T + 4.03e7iT^{2} \)
11 \( 1 + (4.09e4 + 4.09e4i)T + 2.35e9iT^{2} \)
13 \( 1 + 5.07e4T + 1.06e10T^{2} \)
19 \( 1 + 8.15e5iT - 3.22e11T^{2} \)
23 \( 1 + (-5.26e5 - 5.26e5i)T + 1.80e12iT^{2} \)
29 \( 1 + (7.14e5 - 7.14e5i)T - 1.45e13iT^{2} \)
31 \( 1 + (9.24e5 - 9.24e5i)T - 2.64e13iT^{2} \)
37 \( 1 + (1.11e7 - 1.11e7i)T - 1.29e14iT^{2} \)
41 \( 1 + (3.49e6 + 3.49e6i)T + 3.27e14iT^{2} \)
43 \( 1 - 4.64e6iT - 5.02e14T^{2} \)
47 \( 1 + 3.34e7T + 1.11e15T^{2} \)
53 \( 1 - 5.09e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.72e8iT - 8.66e15T^{2} \)
61 \( 1 + (-3.40e7 - 3.40e7i)T + 1.16e16iT^{2} \)
67 \( 1 - 2.00e8T + 2.72e16T^{2} \)
71 \( 1 + (-7.24e7 + 7.24e7i)T - 4.58e16iT^{2} \)
73 \( 1 + (3.09e8 - 3.09e8i)T - 5.88e16iT^{2} \)
79 \( 1 + (-2.19e8 - 2.19e8i)T + 1.19e17iT^{2} \)
83 \( 1 + 8.30e8iT - 1.86e17T^{2} \)
89 \( 1 - 6.52e7T + 3.50e17T^{2} \)
97 \( 1 + (1.59e8 - 1.59e8i)T - 7.60e17iT^{2} \)
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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

−16.16538549005712896599819718268, −15.47727681652752174664514442492, −13.23049135026218338869377798640, −11.53926985442434171102459478192, −10.82496450629105786739658261491, −9.687580837002456457474936454274, −6.94251664733507309789907612296, −5.15288684214592762249995432505, −3.17958471583028442001626381754, −0.32245834089284252936636536274, 2.02753850411875170551158423161, 5.56436841066251373080330027795, 6.65001182451015761259174371590, 7.898809915289394569433006539860, 10.47522541038434060791734257932, 12.05416405208187322928109367740, 12.73501238040769428548270637957, 14.81589292739805862790971737834, 16.11322525173179349996009346194, 17.17006382265200265865248006687

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