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Properties

Label 2-17e2-17.15-c1-0-7
Degree $2$
Conductor $289$
Sign $0.739 - 0.673i$
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  + (0.292 + 0.292i)2-s + (1 + 2.41i)3-s − 1.82i·4-s + (1.70 − 0.707i)5-s + (−0.414 + i)6-s + (1 + 0.414i)7-s + (1.12 − 1.12i)8-s + (−2.70 + 2.70i)9-s + (0.707 + 0.292i)10-s + (0.414 − i)11-s + (4.41 − 1.82i)12-s + 1.41i·13-s + (0.171 + 0.414i)14-s + (3.41 + 3.41i)15-s − 3·16-s + ⋯
L(s)  = 1  + (0.207 + 0.207i)2-s + (0.577 + 1.39i)3-s − 0.914i·4-s + (0.763 − 0.316i)5-s + (−0.169 + 0.408i)6-s + (0.377 + 0.156i)7-s + (0.396 − 0.396i)8-s + (−0.902 + 0.902i)9-s + (0.223 + 0.0926i)10-s + (0.124 − 0.301i)11-s + (1.27 − 0.527i)12-s + 0.392i·13-s + (0.0458 + 0.110i)14-s + (0.881 + 0.881i)15-s − 0.750·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.739 - 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.739 - 0.673i$
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: \((2,\ 289,\ (\ :1/2),\ 0.739 - 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.77754 + 0.688285i\)
\(L(\frac12)\) \(\approx\) \(1.77754 + 0.688285i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.292 - 0.292i)T + 2iT^{2} \)
3 \( 1 + (-1 - 2.41i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.70 + 0.707i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1 - 0.414i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-0.414 + i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 1.41iT - 13T^{2} \)
19 \( 1 + (3.41 + 3.41i)T + 19iT^{2} \)
23 \( 1 + (1.58 - 3.82i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (4.12 - 1.70i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.24 - 3i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.46 - 3.53i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (7.53 + 3.12i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-3.41 + 3.41i)T - 43iT^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 + (-1 - i)T + 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (-8.53 - 3.53i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 6.82T + 67T^{2} \)
71 \( 1 + (5 + 12.0i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (4.94 - 2.05i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.58 + 3.82i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-0.242 - 0.242i)T + 83iT^{2} \)
89 \( 1 - 9.41iT - 89T^{2} \)
97 \( 1 + (-5.94 + 2.46i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.61274342635168675416028077263, −10.68766221136448263853435092929, −9.935909462059240708052846304017, −9.242095146683469175467397035372, −8.538763052765980488156974061602, −6.80811284743414361518353025453, −5.53782225296364853650503766152, −4.90601675511656492646430717855, −3.75716048981356610768647478481, −1.97350440518711287984089020473, 1.85549633183620166312671814418, 2.72128596201450061562416938978, 4.22346247681032732201135755280, 5.98717967161169623508860311342, 6.94418421593274508239845283602, 7.898714884179460876652557394896, 8.414196852469738596704383532796, 9.755176925509473463560211328363, 10.97912637677818324969061389649, 12.02684394907266479597696161734

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