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Properties

Label 2-17e2-17.15-c1-0-1
Degree $2$
Conductor $289$
Sign $0.632 - 0.774i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.921 − 0.921i)2-s + (0.881 + 2.12i)3-s − 0.302i·4-s + (−1.20 + 0.498i)5-s + (1.14 − 2.77i)6-s + (3.05 + 1.26i)7-s + (−2.12 + 2.12i)8-s + (−1.62 + 1.62i)9-s + (1.56 + 0.649i)10-s + (−1.14 + 2.77i)11-s + (0.644 − 0.266i)12-s − 0.302i·13-s + (−1.64 − 3.97i)14-s + (−2.12 − 2.12i)15-s + 3.30·16-s + ⋯
L(s)  = 1  + (−0.651 − 0.651i)2-s + (0.508 + 1.22i)3-s − 0.151i·4-s + (−0.538 + 0.222i)5-s + (0.468 − 1.13i)6-s + (1.15 + 0.477i)7-s + (−0.749 + 0.749i)8-s + (−0.542 + 0.542i)9-s + (0.495 + 0.205i)10-s + (−0.346 + 0.835i)11-s + (0.185 − 0.0770i)12-s − 0.0839i·13-s + (−0.440 − 1.06i)14-s + (−0.547 − 0.547i)15-s + 0.825·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.632 - 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.632 - 0.774i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.632 - 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.923497 + 0.438260i\)
\(L(\frac12)\) \(\approx\) \(0.923497 + 0.438260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (0.921 + 0.921i)T + 2iT^{2} \)
3 \( 1 + (-0.881 - 2.12i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (1.20 - 0.498i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-3.05 - 1.26i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.14 - 2.77i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.302iT - 13T^{2} \)
19 \( 1 + (-3.47 - 3.47i)T + 19iT^{2} \)
23 \( 1 + (0.498 - 1.20i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.839 + 0.347i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.37 - 3.33i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (2.52 + 6.10i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (-5.54 - 2.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-6.79 + 6.79i)T - 43iT^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + (9.12 + 9.12i)T + 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (11.8 + 4.90i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.39T + 67T^{2} \)
71 \( 1 + (-4.29 - 10.3i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.02 + 2.91i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.22 + 2.96i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.9 + 10.9i)T + 83iT^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (-0.0846 + 0.0350i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.56629971731304705267488130903, −10.82916397386202690957450090357, −10.03765244154968628731830481428, −9.327172827930079791601800465771, −8.435003605254906093199458881415, −7.54637780984625748517517578512, −5.56912814134641222871698928118, −4.66736897782669621137906511901, −3.37026228168423605126365770112, −1.93448051759604241637718256395, 0.975274535649210796549081296332, 2.85422869254762189123007518972, 4.45305979676054369896265019345, 6.13086728309993923461346939185, 7.26989040776070068009024822087, 7.88396322793739195137915957831, 8.278148272512134631126609421103, 9.302432285002971021829516508533, 10.86898007330964972063807474924, 11.80025875551702206085650251899

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