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Properties

Label 2-17e2-17.15-c1-0-2
Degree $2$
Conductor $289$
Sign $-0.722 - 0.691i$
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  + (1.62 + 1.62i)2-s + (0.498 + 1.20i)3-s + 3.30i·4-s + (−2.12 + 0.881i)5-s + (−1.14 + 2.77i)6-s + (0.279 + 0.115i)7-s + (−2.12 + 2.12i)8-s + (0.921 − 0.921i)9-s + (−4.89 − 2.02i)10-s + (1.14 − 2.77i)11-s + (−3.97 + 1.64i)12-s + 3.30i·13-s + (0.266 + 0.644i)14-s + (−2.12 − 2.12i)15-s − 0.302·16-s + ⋯
L(s)  = 1  + (1.15 + 1.15i)2-s + (0.287 + 0.694i)3-s + 1.65i·4-s + (−0.951 + 0.394i)5-s + (−0.468 + 1.13i)6-s + (0.105 + 0.0437i)7-s + (−0.750 + 0.750i)8-s + (0.307 − 0.307i)9-s + (−1.54 − 0.641i)10-s + (0.346 − 0.835i)11-s + (−1.14 + 0.475i)12-s + 0.916i·13-s + (0.0713 + 0.172i)14-s + (−0.547 − 0.547i)15-s − 0.0756·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.722 - 0.691i$
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: \((2,\ 289,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.832666 + 2.07358i\)
\(L(\frac12)\) \(\approx\) \(0.832666 + 2.07358i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.62 - 1.62i)T + 2iT^{2} \)
3 \( 1 + (-0.498 - 1.20i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (2.12 - 0.881i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.279 - 0.115i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.14 + 2.77i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.30iT - 13T^{2} \)
19 \( 1 + (4.17 + 4.17i)T + 19iT^{2} \)
23 \( 1 + (0.881 - 2.12i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-9.15 + 3.79i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-1.37 - 3.33i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (0.231 + 0.559i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.54 + 2.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-1.69 + 1.69i)T - 43iT^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + (1.47 + 1.47i)T + 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (8.14 + 3.37i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 12.6T + 67T^{2} \)
71 \( 1 + (-1.22 - 2.96i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (0.364 - 0.150i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.29 + 10.3i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-1.77 - 1.77i)T + 83iT^{2} \)
89 \( 1 - 3.21iT - 89T^{2} \)
97 \( 1 + (10.0 - 4.17i)T + (68.5 - 68.5i)T^{2} \)
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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

−12.21975998041577558259498376464, −11.51482468963546453600774488029, −10.36026700640453562843544573995, −8.986445697453161972911609443092, −8.148543265767403302456431545713, −6.95250227525898813340364214462, −6.37164679107132752359840928296, −4.84359977043694298934299278877, −4.08678681307396725436975255406, −3.26920933717309427453091557127, 1.43752175284190436220045250719, 2.77809753252972793209110002051, 4.14847937801546192286729377026, 4.79028467672669967557272476784, 6.31120187411816360364782337210, 7.67774310079513690480193134189, 8.371261164288576550060983969771, 10.06781754229652675000037380497, 10.71208188382698722523388288993, 11.95191346453216734929934951620

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