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Properties

Label 2-17-17.13-c9-0-3
Degree $2$
Conductor $17$
Sign $0.661 - 0.749i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7.42i·2-s + (150. + 150. i)3-s + 456.·4-s + (210. + 210. i)5-s + (1.12e3 − 1.12e3i)6-s + (−5.46e3 + 5.46e3i)7-s − 7.19e3i·8-s + 2.58e4i·9-s + (1.56e3 − 1.56e3i)10-s + (6.56e3 − 6.56e3i)11-s + (6.89e4 + 6.89e4i)12-s + 1.22e5·13-s + (4.05e4 + 4.05e4i)14-s + 6.36e4i·15-s + 1.80e5·16-s + (−3.43e5 − 2.07e4i)17-s + ⋯
L(s)  = 1  − 0.328i·2-s + (1.07 + 1.07i)3-s + 0.892·4-s + (0.150 + 0.150i)5-s + (0.353 − 0.353i)6-s + (−0.859 + 0.859i)7-s − 0.621i·8-s + 1.31i·9-s + (0.0495 − 0.0495i)10-s + (0.135 − 0.135i)11-s + (0.959 + 0.959i)12-s + 1.18·13-s + (0.282 + 0.282i)14-s + 0.324i·15-s + 0.688·16-s + (−0.998 − 0.0602i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.661 - 0.749i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 0.661 - 0.749i)\)

Particular Values

\(L(5)\) \(\approx\) \(2.48778 + 1.12230i\)
\(L(\frac12)\) \(\approx\) \(2.48778 + 1.12230i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (3.43e5 + 2.07e4i)T \)
good2 \( 1 + 7.42iT - 512T^{2} \)
3 \( 1 + (-150. - 150. i)T + 1.96e4iT^{2} \)
5 \( 1 + (-210. - 210. i)T + 1.95e6iT^{2} \)
7 \( 1 + (5.46e3 - 5.46e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (-6.56e3 + 6.56e3i)T - 2.35e9iT^{2} \)
13 \( 1 - 1.22e5T + 1.06e10T^{2} \)
19 \( 1 - 4.64e5iT - 3.22e11T^{2} \)
23 \( 1 + (-8.29e5 + 8.29e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (-9.29e5 - 9.29e5i)T + 1.45e13iT^{2} \)
31 \( 1 + (5.27e6 + 5.27e6i)T + 2.64e13iT^{2} \)
37 \( 1 + (-1.11e6 - 1.11e6i)T + 1.29e14iT^{2} \)
41 \( 1 + (-1.72e7 + 1.72e7i)T - 3.27e14iT^{2} \)
43 \( 1 - 7.29e6iT - 5.02e14T^{2} \)
47 \( 1 + 5.46e7T + 1.11e15T^{2} \)
53 \( 1 + 6.10e7iT - 3.29e15T^{2} \)
59 \( 1 + 8.90e7iT - 8.66e15T^{2} \)
61 \( 1 + (-1.86e7 + 1.86e7i)T - 1.16e16iT^{2} \)
67 \( 1 + 2.68e8T + 2.72e16T^{2} \)
71 \( 1 + (-2.78e8 - 2.78e8i)T + 4.58e16iT^{2} \)
73 \( 1 + (-1.98e8 - 1.98e8i)T + 5.88e16iT^{2} \)
79 \( 1 + (4.29e8 - 4.29e8i)T - 1.19e17iT^{2} \)
83 \( 1 + 5.22e8iT - 1.86e17T^{2} \)
89 \( 1 + 2.71e8T + 3.50e17T^{2} \)
97 \( 1 + (-6.26e8 - 6.26e8i)T + 7.60e17iT^{2} \)
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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

−16.24550443895083760068415927161, −15.67868734869095791806669609220, −14.54245889054535507171520210595, −12.85309782641945418155496928081, −11.08331347595087022063941803531, −9.797247141110453601108407968478, −8.581406180978741063216337263330, −6.28711859059264266603797774101, −3.65635920308389954255164749928, −2.42646686996479529354670166902, 1.44672373515813030236385759672, 3.16024844810567875072016130568, 6.51018041404519693901889591484, 7.41478179131122396600767166014, 8.969207012338431044784316879102, 11.03638280125250745337842467454, 12.93370851680232656313128102803, 13.67958798030467678698704758513, 15.17887693669835731023489215927, 16.44032013663811528171606013273

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