Properties

Label 2-17e2-17.13-c1-0-7
Degree $2$
Conductor $289$
Sign $0.992 + 0.122i$
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  + 2.30i·2-s + (−0.921 − 0.921i)3-s − 3.30·4-s + (−1.62 − 1.62i)5-s + (2.12 − 2.12i)6-s + (0.214 − 0.214i)7-s − 3.00i·8-s − 1.30i·9-s + (3.74 − 3.74i)10-s + (2.12 − 2.12i)11-s + (3.04 + 3.04i)12-s + 3.30·13-s + (0.493 + 0.493i)14-s + 3i·15-s + 0.302·16-s + ⋯
L(s)  = 1  + 1.62i·2-s + (−0.531 − 0.531i)3-s − 1.65·4-s + (−0.728 − 0.728i)5-s + (0.866 − 0.866i)6-s + (0.0809 − 0.0809i)7-s − 1.06i·8-s − 0.434i·9-s + (1.18 − 1.18i)10-s + (0.639 − 0.639i)11-s + (0.878 + 0.878i)12-s + 0.916·13-s + (0.131 + 0.131i)14-s + 0.774i·15-s + 0.0756·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.776041 - 0.0475879i\)
\(L(\frac12)\) \(\approx\) \(0.776041 - 0.0475879i\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.30iT - 2T^{2} \)
3 \( 1 + (0.921 + 0.921i)T + 3iT^{2} \)
5 \( 1 + (1.62 + 1.62i)T + 5iT^{2} \)
7 \( 1 + (-0.214 + 0.214i)T - 7iT^{2} \)
11 \( 1 + (-2.12 + 2.12i)T - 11iT^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
19 \( 1 + 5.90iT - 19T^{2} \)
23 \( 1 + (-1.62 + 1.62i)T - 23iT^{2} \)
29 \( 1 + (7.00 + 7.00i)T + 29iT^{2} \)
31 \( 1 + (-2.54 - 2.54i)T + 31iT^{2} \)
37 \( 1 + (-0.428 - 0.428i)T + 37iT^{2} \)
41 \( 1 + (4.24 - 4.24i)T - 41iT^{2} \)
43 \( 1 + 2.39iT - 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 2.09iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (-6.23 + 6.23i)T - 61iT^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + (2.27 + 2.27i)T + 71iT^{2} \)
73 \( 1 + (0.278 + 0.278i)T + 73iT^{2} \)
79 \( 1 + (-7.92 + 7.92i)T - 79iT^{2} \)
83 \( 1 + 2.51iT - 83T^{2} \)
89 \( 1 + 3.21T + 89T^{2} \)
97 \( 1 + (-7.71 - 7.71i)T + 97iT^{2} \)
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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

−11.78853995596079153925759031561, −11.18918452866924144060568263700, −9.298664642189460568827923653355, −8.632708390699628781774367604389, −7.79452130087395223949628832233, −6.72271495860641107291319298142, −6.13185587054695289610042114041, −4.99166438260935372295329067012, −3.87438777188536064300634592030, −0.66469491669643164941383628440, 1.77431148468465594407927335305, 3.45531144562811560230955273727, 4.08995802411882154002554354026, 5.41203026112395116941722145693, 6.96041850108825243479115703198, 8.258906710180038090128875705642, 9.495252297284421128755928071754, 10.29444590449976926502907510242, 11.08826561699529803781413280083, 11.46705168899192555656767006317

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