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Properties

Label 2-17-17.4-c9-0-10
Degree $2$
Conductor $17$
Sign $-0.991 - 0.131i$
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  − 31.4i·2-s + (42.7 − 42.7i)3-s − 474.·4-s + (1.11e3 − 1.11e3i)5-s + (−1.34e3 − 1.34e3i)6-s + (−7.41e3 − 7.41e3i)7-s − 1.16e3i·8-s + 1.60e4i·9-s + (−3.51e4 − 3.51e4i)10-s + (5.52e4 + 5.52e4i)11-s + (−2.02e4 + 2.02e4i)12-s − 1.11e5·13-s + (−2.32e5 + 2.32e5i)14-s − 9.56e4i·15-s − 2.79e5·16-s + (2.45e5 − 2.41e5i)17-s + ⋯
L(s)  = 1  − 1.38i·2-s + (0.304 − 0.304i)3-s − 0.927·4-s + (0.800 − 0.800i)5-s + (−0.422 − 0.422i)6-s + (−1.16 − 1.16i)7-s − 0.100i·8-s + 0.814i·9-s + (−1.11 − 1.11i)10-s + (1.13 + 1.13i)11-s + (−0.282 + 0.282i)12-s − 1.08·13-s + (−1.62 + 1.62i)14-s − 0.487i·15-s − 1.06·16-s + (0.713 − 0.700i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.131i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.991 - 0.131i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.991 - 0.131i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.115412 + 1.75380i\)
\(L(\frac12)\) \(\approx\) \(0.115412 + 1.75380i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-2.45e5 + 2.41e5i)T \)
good2 \( 1 + 31.4iT - 512T^{2} \)
3 \( 1 + (-42.7 + 42.7i)T - 1.96e4iT^{2} \)
5 \( 1 + (-1.11e3 + 1.11e3i)T - 1.95e6iT^{2} \)
7 \( 1 + (7.41e3 + 7.41e3i)T + 4.03e7iT^{2} \)
11 \( 1 + (-5.52e4 - 5.52e4i)T + 2.35e9iT^{2} \)
13 \( 1 + 1.11e5T + 1.06e10T^{2} \)
19 \( 1 + 2.02e5iT - 3.22e11T^{2} \)
23 \( 1 + (5.36e5 + 5.36e5i)T + 1.80e12iT^{2} \)
29 \( 1 + (-5.29e6 + 5.29e6i)T - 1.45e13iT^{2} \)
31 \( 1 + (-7.51e5 + 7.51e5i)T - 2.64e13iT^{2} \)
37 \( 1 + (-7.95e6 + 7.95e6i)T - 1.29e14iT^{2} \)
41 \( 1 + (9.28e5 + 9.28e5i)T + 3.27e14iT^{2} \)
43 \( 1 + 5.13e6iT - 5.02e14T^{2} \)
47 \( 1 - 2.33e7T + 1.11e15T^{2} \)
53 \( 1 + 4.46e7iT - 3.29e15T^{2} \)
59 \( 1 - 1.89e7iT - 8.66e15T^{2} \)
61 \( 1 + (-9.54e7 - 9.54e7i)T + 1.16e16iT^{2} \)
67 \( 1 + 3.24e6T + 2.72e16T^{2} \)
71 \( 1 + (1.71e8 - 1.71e8i)T - 4.58e16iT^{2} \)
73 \( 1 + (-8.06e7 + 8.06e7i)T - 5.88e16iT^{2} \)
79 \( 1 + (3.01e7 + 3.01e7i)T + 1.19e17iT^{2} \)
83 \( 1 - 4.37e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.60e7T + 3.50e17T^{2} \)
97 \( 1 + (1.07e9 - 1.07e9i)T - 7.60e17iT^{2} \)
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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

−16.42876625427079039556359940677, −13.98610720617665432124950738553, −13.06913959049464464554502035759, −12.10639518593940346051194116755, −10.10017150412235182047450356614, −9.547489157662305496777420000178, −7.05781706203949148713566909223, −4.38430952674280068855175368856, −2.38799056645863859172787496625, −0.871968775128052950505961159140, 3.03824601506700611598835919213, 5.92752968694465099848661654179, 6.56181907378666241354016504204, 8.740736071856755934961460366502, 9.818101967534220149441304406663, 12.16287326102952832446526028812, 14.15194803369175646805583028932, 14.79802584664629503302944377390, 15.97852350331764339772984357930, 17.12218368728271094277843685294

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