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Properties

Label 2-17-17.2-c3-0-1
Degree $2$
Conductor $17$
Sign $0.713 + 0.700i$
Analytic cond. $1.00303$
Root an. cond. $1.00151$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.20 − 1.20i)2-s + (4.10 − 1.70i)3-s − 5.09i·4-s + (2.60 + 6.29i)5-s + (−7.00 − 2.90i)6-s + (−5.31 + 12.8i)7-s + (−15.7 + 15.7i)8-s + (−5.12 + 5.12i)9-s + (4.44 − 10.7i)10-s + (28.4 + 11.8i)11-s + (−8.65 − 20.9i)12-s − 66.0i·13-s + (21.8 − 9.05i)14-s + (21.3 + 21.3i)15-s − 2.65·16-s + (3.91 + 69.9i)17-s + ⋯
L(s)  = 1  + (−0.426 − 0.426i)2-s + (0.790 − 0.327i)3-s − 0.636i·4-s + (0.233 + 0.562i)5-s + (−0.476 − 0.197i)6-s + (−0.286 + 0.692i)7-s + (−0.697 + 0.697i)8-s + (−0.189 + 0.189i)9-s + (0.140 − 0.339i)10-s + (0.780 + 0.323i)11-s + (−0.208 − 0.502i)12-s − 1.40i·13-s + (0.417 − 0.172i)14-s + (0.368 + 0.368i)15-s − 0.0414·16-s + (0.0558 + 0.998i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(1.00303\)
Root analytic conductor: \(1.00151\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :3/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.939408 - 0.383954i\)
\(L(\frac12)\) \(\approx\) \(0.939408 - 0.383954i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.91 - 69.9i)T \)
good2 \( 1 + (1.20 + 1.20i)T + 8iT^{2} \)
3 \( 1 + (-4.10 + 1.70i)T + (19.0 - 19.0i)T^{2} \)
5 \( 1 + (-2.60 - 6.29i)T + (-88.3 + 88.3i)T^{2} \)
7 \( 1 + (5.31 - 12.8i)T + (-242. - 242. i)T^{2} \)
11 \( 1 + (-28.4 - 11.8i)T + (941. + 941. i)T^{2} \)
13 \( 1 + 66.0iT - 2.19e3T^{2} \)
19 \( 1 + (56.3 + 56.3i)T + 6.85e3iT^{2} \)
23 \( 1 + (26.6 + 11.0i)T + (8.60e3 + 8.60e3i)T^{2} \)
29 \( 1 + (101. + 245. i)T + (-1.72e4 + 1.72e4i)T^{2} \)
31 \( 1 + (33.0 - 13.6i)T + (2.10e4 - 2.10e4i)T^{2} \)
37 \( 1 + (-330. + 136. i)T + (3.58e4 - 3.58e4i)T^{2} \)
41 \( 1 + (-12.5 + 30.2i)T + (-4.87e4 - 4.87e4i)T^{2} \)
43 \( 1 + (364. - 364. i)T - 7.95e4iT^{2} \)
47 \( 1 - 210. iT - 1.03e5T^{2} \)
53 \( 1 + (-61.6 - 61.6i)T + 1.48e5iT^{2} \)
59 \( 1 + (-219. + 219. i)T - 2.05e5iT^{2} \)
61 \( 1 + (-139. + 337. i)T + (-1.60e5 - 1.60e5i)T^{2} \)
67 \( 1 - 660.T + 3.00e5T^{2} \)
71 \( 1 + (367. - 152. i)T + (2.53e5 - 2.53e5i)T^{2} \)
73 \( 1 + (-246. - 594. i)T + (-2.75e5 + 2.75e5i)T^{2} \)
79 \( 1 + (355. + 147. i)T + (3.48e5 + 3.48e5i)T^{2} \)
83 \( 1 + (-108. - 108. i)T + 5.71e5iT^{2} \)
89 \( 1 + 599. iT - 7.04e5T^{2} \)
97 \( 1 + (17.0 + 41.1i)T + (-6.45e5 + 6.45e5i)T^{2} \)
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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

−18.61188605735192498881607575611, −17.43873115178166455887750190833, −15.18418786612221457492019306711, −14.52417165558527927629050291221, −12.92135829999440561470872235501, −11.09076355861067274968795889258, −9.711265958744175956455694483695, −8.327495643475991069037345447574, −6.06269400590765625480868872416, −2.48354463017318552863143116580, 3.79242950438097635350486317070, 6.87717063410500038810814649426, 8.660533138247564279739089166711, 9.463827456949105805607824573335, 11.83612656132196482411474135993, 13.44453564273511096833794945574, 14.66784985093926259598485749841, 16.47114711113462620669420948430, 16.84670541669437573410909048225, 18.52785607936221278184299878936

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