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Properties

Label 2-17e2-17.13-c1-0-2
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

\(L(1)\) \(\approx\) \(0.0934770 + 1.52437i\)
\(L(\frac12)\) \(\approx\) \(0.0934770 + 1.52437i\)
\(L(\frac{3}{2})\) 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

−12.63076081239309002421478670092, −11.03071747906108485563404641612, −10.01728793880721022237416108493, −9.157001298097049004367721320347, −8.426634808319124017437193139408, −7.18273729863877035317666098353, −6.48669062407870450320134996483, −5.51058183254345649374564181865, −4.30948009801872633055600677197, −2.77580036434225984197944275099, 1.26093247842028025423191443160, 2.32444923179723448596170807849, 3.54379623807511000344881312983, 4.93228060946024746164968740632, 6.17612648945609684611300849021, 8.024322955217820217103986219206, 8.620760392662429654318902419755, 9.718984689157282687923169917724, 10.46167644503565433427265492504, 11.32521595247574414500294419310

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