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Properties

Label 2-17e2-17.13-c1-0-5
Degree $2$
Conductor $289$
Sign $0.901 + 0.432i$
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  + 0.414i·2-s + (−1.84 − 1.84i)3-s + 1.82·4-s + (1.30 + 1.30i)5-s + (0.765 − 0.765i)6-s + (0.765 − 0.765i)7-s + 1.58i·8-s + 3.82i·9-s + (−0.541 + 0.541i)10-s + (0.765 − 0.765i)11-s + (−3.37 − 3.37i)12-s + 1.41·13-s + (0.317 + 0.317i)14-s − 4.82i·15-s + 3·16-s + ⋯
L(s)  = 1  + 0.292i·2-s + (−1.06 − 1.06i)3-s + 0.914·4-s + (0.584 + 0.584i)5-s + (0.312 − 0.312i)6-s + (0.289 − 0.289i)7-s + 0.560i·8-s + 1.27i·9-s + (−0.171 + 0.171i)10-s + (0.230 − 0.230i)11-s + (−0.975 − 0.975i)12-s + 0.392·13-s + (0.0847 + 0.0847i)14-s − 1.24i·15-s + 0.750·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.901 + 0.432i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.901 + 0.432i$
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: \((2,\ 289,\ (\ :1/2),\ 0.901 + 0.432i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27015 - 0.288674i\)
\(L(\frac12)\) \(\approx\) \(1.27015 - 0.288674i\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.414iT - 2T^{2} \)
3 \( 1 + (1.84 + 1.84i)T + 3iT^{2} \)
5 \( 1 + (-1.30 - 1.30i)T + 5iT^{2} \)
7 \( 1 + (-0.765 + 0.765i)T - 7iT^{2} \)
11 \( 1 + (-0.765 + 0.765i)T - 11iT^{2} \)
13 \( 1 - 1.41T + 13T^{2} \)
19 \( 1 + 4.82iT - 19T^{2} \)
23 \( 1 + (-2.93 + 2.93i)T - 23iT^{2} \)
29 \( 1 + (-3.15 - 3.15i)T + 29iT^{2} \)
31 \( 1 + (-2.29 - 2.29i)T + 31iT^{2} \)
37 \( 1 + (2.70 + 2.70i)T + 37iT^{2} \)
41 \( 1 + (5.76 - 5.76i)T - 41iT^{2} \)
43 \( 1 + 4.82iT - 43T^{2} \)
47 \( 1 + 10.8T + 47T^{2} \)
53 \( 1 - 1.41iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (6.53 - 6.53i)T - 61iT^{2} \)
67 \( 1 + 6.82T + 67T^{2} \)
71 \( 1 + (-9.23 - 9.23i)T + 71iT^{2} \)
73 \( 1 + (3.78 + 3.78i)T + 73iT^{2} \)
79 \( 1 + (-2.93 + 2.93i)T - 79iT^{2} \)
83 \( 1 + 0.343iT - 83T^{2} \)
89 \( 1 + 9.41T + 89T^{2} \)
97 \( 1 + (4.55 + 4.55i)T + 97iT^{2} \)
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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

−11.63890513575525930797834743927, −10.98719216112467237057019202649, −10.31918877753771292438180277458, −8.619529459615059809037468491121, −7.38809371061849324610309855599, −6.66343010082049988758827209399, −6.19498358052239449597655000628, −5.03812911866289339728221104248, −2.79505963918029664180681095437, −1.37382657666514081550385711086, 1.63162276623333711169634711212, 3.54116415336237961114641433409, 4.89997935681123598356019102567, 5.73786882847827848739033586127, 6.59539804018171449461196767817, 8.126305371894290139453404901515, 9.499401713283029634824447132289, 10.09625337586062647953829178106, 10.99951741573745775553460763014, 11.69430912024480398526539640413

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