Properties

Label 2-17-17.2-c9-0-8
Degree $2$
Conductor $17$
Sign $0.953 - 0.299i$
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  + (13.0 + 13.0i)2-s + (116. − 48.3i)3-s − 170. i·4-s + (663. + 1.60e3i)5-s + (2.15e3 + 893. i)6-s + (2.94e3 − 7.11e3i)7-s + (8.91e3 − 8.91e3i)8-s + (−2.62e3 + 2.62e3i)9-s + (−1.22e4 + 2.95e4i)10-s + (5.73e4 + 2.37e4i)11-s + (−8.26e3 − 1.99e4i)12-s + 7.04e4i·13-s + (1.31e5 − 5.44e4i)14-s + (1.54e5 + 1.54e5i)15-s + 1.45e5·16-s + (−3.12e5 − 1.44e5i)17-s + ⋯
L(s)  = 1  + (0.577 + 0.577i)2-s + (0.832 − 0.344i)3-s − 0.333i·4-s + (0.474 + 1.14i)5-s + (0.679 + 0.281i)6-s + (0.463 − 1.11i)7-s + (0.769 − 0.769i)8-s + (−0.133 + 0.133i)9-s + (−0.387 + 0.934i)10-s + (1.18 + 0.488i)11-s + (−0.115 − 0.277i)12-s + 0.684i·13-s + (0.913 − 0.378i)14-s + (0.789 + 0.789i)15-s + 0.554·16-s + (−0.907 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.953 - 0.299i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.953 - 0.299i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.953 - 0.299i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 0.953 - 0.299i)\)

Particular Values

\(L(5)\) \(\approx\) \(3.20633 + 0.491998i\)
\(L(\frac12)\) \(\approx\) \(3.20633 + 0.491998i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.12e5 + 1.44e5i)T \)
good2 \( 1 + (-13.0 - 13.0i)T + 512iT^{2} \)
3 \( 1 + (-116. + 48.3i)T + (1.39e4 - 1.39e4i)T^{2} \)
5 \( 1 + (-663. - 1.60e3i)T + (-1.38e6 + 1.38e6i)T^{2} \)
7 \( 1 + (-2.94e3 + 7.11e3i)T + (-2.85e7 - 2.85e7i)T^{2} \)
11 \( 1 + (-5.73e4 - 2.37e4i)T + (1.66e9 + 1.66e9i)T^{2} \)
13 \( 1 - 7.04e4iT - 1.06e10T^{2} \)
19 \( 1 + (-2.16e5 - 2.16e5i)T + 3.22e11iT^{2} \)
23 \( 1 + (1.51e6 + 6.28e5i)T + (1.27e12 + 1.27e12i)T^{2} \)
29 \( 1 + (2.59e6 + 6.26e6i)T + (-1.02e13 + 1.02e13i)T^{2} \)
31 \( 1 + (5.82e6 - 2.41e6i)T + (1.86e13 - 1.86e13i)T^{2} \)
37 \( 1 + (-6.43e6 + 2.66e6i)T + (9.18e13 - 9.18e13i)T^{2} \)
41 \( 1 + (2.52e6 - 6.09e6i)T + (-2.31e14 - 2.31e14i)T^{2} \)
43 \( 1 + (1.82e7 - 1.82e7i)T - 5.02e14iT^{2} \)
47 \( 1 - 2.56e7iT - 1.11e15T^{2} \)
53 \( 1 + (-8.36e6 - 8.36e6i)T + 3.29e15iT^{2} \)
59 \( 1 + (-6.76e7 + 6.76e7i)T - 8.66e15iT^{2} \)
61 \( 1 + (5.29e7 - 1.27e8i)T + (-8.26e15 - 8.26e15i)T^{2} \)
67 \( 1 + 5.84e7T + 2.72e16T^{2} \)
71 \( 1 + (-2.81e8 + 1.16e8i)T + (3.24e16 - 3.24e16i)T^{2} \)
73 \( 1 + (2.76e7 + 6.68e7i)T + (-4.16e16 + 4.16e16i)T^{2} \)
79 \( 1 + (-4.09e8 - 1.69e8i)T + (8.47e16 + 8.47e16i)T^{2} \)
83 \( 1 + (2.13e8 + 2.13e8i)T + 1.86e17iT^{2} \)
89 \( 1 + 3.42e7iT - 3.50e17T^{2} \)
97 \( 1 + (9.00e7 + 2.17e8i)T + (-5.37e17 + 5.37e17i)T^{2} \)
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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

−16.65805748027750576984205601406, −14.80805171987269613800734954871, −14.19786443272262237015978202715, −13.59170587806090775575017514385, −11.11733482647262062790085149234, −9.689047579894640464343758508334, −7.46567941466304007674622415096, −6.44128225158587501356705781304, −4.13023372805768446013304679263, −1.86555326773980229728644241842, 1.94909935672930661308391524339, 3.67425931092446268715858169318, 5.38861842836829245493647357890, 8.460126217332300996369618242638, 9.127008164623230443212093221400, 11.51281512870775324654160800317, 12.61434017970231380586257462050, 13.81200684077350989303853839589, 15.05877111092964607665468980112, 16.65317514174181305999006590699

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