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Properties

Label 2-17-17.13-c9-0-10
Degree $2$
Conductor $17$
Sign $0.0125 - 0.999i$
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  − 28.8i·2-s + (−179. − 179. i)3-s − 319.·4-s + (−1.13e3 − 1.13e3i)5-s + (−5.17e3 + 5.17e3i)6-s + (4.92e3 − 4.92e3i)7-s − 5.55e3i·8-s + 4.46e4i·9-s + (−3.26e4 + 3.26e4i)10-s + (4.48e4 − 4.48e4i)11-s + (5.72e4 + 5.72e4i)12-s + 1.14e5·13-s + (−1.41e5 − 1.41e5i)14-s + 4.06e5i·15-s − 3.23e5·16-s + (−2.74e5 + 2.08e5i)17-s + ⋯
L(s)  = 1  − 1.27i·2-s + (−1.27 − 1.27i)3-s − 0.623·4-s + (−0.810 − 0.810i)5-s + (−1.62 + 1.62i)6-s + (0.774 − 0.774i)7-s − 0.479i·8-s + 2.26i·9-s + (−1.03 + 1.03i)10-s + (0.922 − 0.922i)11-s + (0.797 + 0.797i)12-s + 1.11·13-s + (−0.987 − 0.987i)14-s + 2.07i·15-s − 1.23·16-s + (−0.795 + 0.605i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0125 - 0.999i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.0125 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.0125 - 0.999i$
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: \((2,\ 17,\ (\ :9/2),\ 0.0125 - 0.999i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.681029 + 0.672543i\)
\(L(\frac12)\) \(\approx\) \(0.681029 + 0.672543i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (2.74e5 - 2.08e5i)T \)
good2 \( 1 + 28.8iT - 512T^{2} \)
3 \( 1 + (179. + 179. i)T + 1.96e4iT^{2} \)
5 \( 1 + (1.13e3 + 1.13e3i)T + 1.95e6iT^{2} \)
7 \( 1 + (-4.92e3 + 4.92e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (-4.48e4 + 4.48e4i)T - 2.35e9iT^{2} \)
13 \( 1 - 1.14e5T + 1.06e10T^{2} \)
19 \( 1 + 5.04e4iT - 3.22e11T^{2} \)
23 \( 1 + (-8.05e5 + 8.05e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (1.44e6 + 1.44e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-7.34e5 - 7.34e5i)T + 2.64e13iT^{2} \)
37 \( 1 + (-1.30e7 - 1.30e7i)T + 1.29e14iT^{2} \)
41 \( 1 + (-5.63e6 + 5.63e6i)T - 3.27e14iT^{2} \)
43 \( 1 - 2.09e7iT - 5.02e14T^{2} \)
47 \( 1 - 6.43e6T + 1.11e15T^{2} \)
53 \( 1 + 1.61e7iT - 3.29e15T^{2} \)
59 \( 1 + 1.17e8iT - 8.66e15T^{2} \)
61 \( 1 + (-2.49e7 + 2.49e7i)T - 1.16e16iT^{2} \)
67 \( 1 - 2.32e8T + 2.72e16T^{2} \)
71 \( 1 + (5.09e6 + 5.09e6i)T + 4.58e16iT^{2} \)
73 \( 1 + (2.23e8 + 2.23e8i)T + 5.88e16iT^{2} \)
79 \( 1 + (4.12e8 - 4.12e8i)T - 1.19e17iT^{2} \)
83 \( 1 - 2.61e8iT - 1.86e17T^{2} \)
89 \( 1 - 8.28e7T + 3.50e17T^{2} \)
97 \( 1 + (1.17e8 + 1.17e8i)T + 7.60e17iT^{2} \)
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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

−16.26719676491372733515466100351, −13.53400665174391592499394901227, −12.58952062614773575226413490175, −11.33146441065036420279422818020, −11.11227965685276522536451185310, −8.286454382581982567935270006334, −6.50857049593334867750995653383, −4.29096149861413812627558369446, −1.34182021302905190296934215217, −0.65483522192334600380364905640, 4.27258731875919861470908812294, 5.68047235036590176102484777912, 7.02462100023098825458031073413, 9.041113519581798807410435216707, 11.09766646320789574504403448338, 11.63942762994676364084434565117, 14.73273660497733910878266895360, 15.31237859169478937662345928963, 16.12416225438007764288727880668, 17.39655881728807216299553807247

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