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Properties

Label 2-17e2-17.15-c1-0-11
Degree $2$
Conductor $289$
Sign $-0.00904 + 0.999i$
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  + (−0.245 − 0.245i)2-s + (−0.336 − 0.812i)3-s − 1.87i·4-s + (2.16 − 0.898i)5-s + (−0.116 + 0.282i)6-s + (1.73 + 0.719i)7-s + (−0.952 + 0.952i)8-s + (1.57 − 1.57i)9-s + (−0.753 − 0.311i)10-s + (−1.93 + 4.67i)11-s + (−1.52 + 0.632i)12-s − 4.71i·13-s + (−0.249 − 0.603i)14-s + (−1.45 − 1.45i)15-s − 3.29·16-s + ⋯
L(s)  = 1  + (−0.173 − 0.173i)2-s + (−0.194 − 0.469i)3-s − 0.939i·4-s + (0.969 − 0.401i)5-s + (−0.0477 + 0.115i)6-s + (0.656 + 0.271i)7-s + (−0.336 + 0.336i)8-s + (0.524 − 0.524i)9-s + (−0.238 − 0.0986i)10-s + (−0.584 + 1.41i)11-s + (−0.440 + 0.182i)12-s − 1.30i·13-s + (−0.0667 − 0.161i)14-s + (−0.376 − 0.376i)15-s − 0.822·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00904 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00904 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.00904 + 0.999i$
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: \((2,\ 289,\ (\ :1/2),\ -0.00904 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.934023 - 0.942508i\)
\(L(\frac12)\) \(\approx\) \(0.934023 - 0.942508i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.245 + 0.245i)T + 2iT^{2} \)
3 \( 1 + (0.336 + 0.812i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-2.16 + 0.898i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-1.73 - 0.719i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (1.93 - 4.67i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 4.71iT - 13T^{2} \)
19 \( 1 + (0.245 + 0.245i)T + 19iT^{2} \)
23 \( 1 + (-0.678 + 1.63i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-2.05 + 0.852i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-0.743 - 1.79i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.36 - 5.70i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.77 + 1.97i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.04 + 1.04i)T - 43iT^{2} \)
47 \( 1 - 8.53iT - 47T^{2} \)
53 \( 1 + (-7.39 - 7.39i)T + 53iT^{2} \)
59 \( 1 + (-3.54 + 3.54i)T - 59iT^{2} \)
61 \( 1 + (0.170 + 0.0707i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 2.44T + 67T^{2} \)
71 \( 1 + (-3.79 - 9.16i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (10.0 - 4.17i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-1.69 + 4.09i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-9.60 - 9.60i)T + 83iT^{2} \)
89 \( 1 + 6.32iT - 89T^{2} \)
97 \( 1 + (-8.57 + 3.54i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.59075237650702960956057895952, −10.26083074408537688187687350303, −9.984050797215429312263396516188, −8.904100437072816831612368609457, −7.66313930429860789790596773730, −6.49070374610362001662907487787, −5.49653213282365441783013638138, −4.74930629955540719297187592999, −2.33096984157331014651358507192, −1.23718706363252605229051616206, 2.18784212915186352652852881708, 3.70600946315297661057833276474, 4.90338863034616528095410332383, 6.14712592783120518937721279316, 7.24016296079913996537459992011, 8.232821136801525209623893419938, 9.170213214248699428156400038689, 10.21420108858393220020115614425, 11.06963147143734166753658001095, 11.79610756751259584893267944787

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