Properties

Label 2-17-17.6-c2-0-0
Degree $2$
Conductor $17$
Sign $0.909 + 0.416i$
Analytic cond. $0.463216$
Root an. cond. $0.680600$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.63 − 0.675i)2-s + (3.96 − 0.789i)3-s + (−0.624 − 0.624i)4-s + (−4.29 + 6.43i)5-s + (−7.00 − 1.39i)6-s + (−2.27 − 3.40i)7-s + (3.29 + 7.96i)8-s + (6.81 − 2.82i)9-s + (11.3 − 7.58i)10-s + (1.35 − 6.80i)11-s + (−2.97 − 1.98i)12-s + (2.37 − 2.37i)13-s + (1.40 + 7.08i)14-s + (−11.9 + 28.9i)15-s − 11.6i·16-s + (−8.76 − 14.5i)17-s + ⋯
L(s)  = 1  + (−0.815 − 0.337i)2-s + (1.32 − 0.263i)3-s + (−0.156 − 0.156i)4-s + (−0.859 + 1.28i)5-s + (−1.16 − 0.232i)6-s + (−0.324 − 0.486i)7-s + (0.412 + 0.995i)8-s + (0.757 − 0.313i)9-s + (1.13 − 0.758i)10-s + (0.123 − 0.618i)11-s + (−0.247 − 0.165i)12-s + (0.182 − 0.182i)13-s + (0.100 + 0.506i)14-s + (−0.798 + 1.92i)15-s − 0.730i·16-s + (−0.515 − 0.856i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.909 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.909 + 0.416i$
Analytic conductor: \(0.463216\)
Root analytic conductor: \(0.680600\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (6, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :1),\ 0.909 + 0.416i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.665538 - 0.145246i\)
\(L(\frac12)\) \(\approx\) \(0.665538 - 0.145246i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (8.76 + 14.5i)T \)
good2 \( 1 + (1.63 + 0.675i)T + (2.82 + 2.82i)T^{2} \)
3 \( 1 + (-3.96 + 0.789i)T + (8.31 - 3.44i)T^{2} \)
5 \( 1 + (4.29 - 6.43i)T + (-9.56 - 23.0i)T^{2} \)
7 \( 1 + (2.27 + 3.40i)T + (-18.7 + 45.2i)T^{2} \)
11 \( 1 + (-1.35 + 6.80i)T + (-111. - 46.3i)T^{2} \)
13 \( 1 + (-2.37 + 2.37i)T - 169iT^{2} \)
19 \( 1 + (-22.7 - 9.43i)T + (255. + 255. i)T^{2} \)
23 \( 1 + (11.2 + 2.24i)T + (488. + 202. i)T^{2} \)
29 \( 1 + (-9.98 - 6.67i)T + (321. + 776. i)T^{2} \)
31 \( 1 + (-7.38 - 37.1i)T + (-887. + 367. i)T^{2} \)
37 \( 1 + (31.6 - 6.29i)T + (1.26e3 - 523. i)T^{2} \)
41 \( 1 + (-18.7 - 28.1i)T + (-643. + 1.55e3i)T^{2} \)
43 \( 1 + (-3.21 + 1.33i)T + (1.30e3 - 1.30e3i)T^{2} \)
47 \( 1 + (-3.16 + 3.16i)T - 2.20e3iT^{2} \)
53 \( 1 + (28.3 + 11.7i)T + (1.98e3 + 1.98e3i)T^{2} \)
59 \( 1 + (4.49 + 10.8i)T + (-2.46e3 + 2.46e3i)T^{2} \)
61 \( 1 + (-58.9 + 39.3i)T + (1.42e3 - 3.43e3i)T^{2} \)
67 \( 1 - 28.5iT - 4.48e3T^{2} \)
71 \( 1 + (33.7 - 6.70i)T + (4.65e3 - 1.92e3i)T^{2} \)
73 \( 1 + (52.0 - 77.8i)T + (-2.03e3 - 4.92e3i)T^{2} \)
79 \( 1 + (-4.18 + 21.0i)T + (-5.76e3 - 2.38e3i)T^{2} \)
83 \( 1 + (-43.4 + 104. i)T + (-4.87e3 - 4.87e3i)T^{2} \)
89 \( 1 + (28.3 + 28.3i)T + 7.92e3iT^{2} \)
97 \( 1 + (138. + 92.4i)T + (3.60e3 + 8.69e3i)T^{2} \)
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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

−18.89920394271429262911958047744, −18.02943907391608344835540231383, −15.92051701593059035196477507523, −14.41872137466452159451442812296, −13.80807738048623079257004034538, −11.36767615429562852365151586144, −10.04004246105681872475083467203, −8.518022812927411086323442658732, −7.30185644386452081727377037083, −3.19692169658130185197820665222, 4.08600961077270380115859267313, 7.74624421637050607435966712129, 8.740907734124751232301824231597, 9.507224013604162901297284423054, 12.25850728512923143788352400893, 13.48204519608541014008585058445, 15.35504793912740461673967334970, 16.08715468415279784044274348760, 17.49106988771116373700500066398, 19.09737596074669099542739045119

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