Properties

Label 2-17-17.15-c9-0-3
Degree $2$
Conductor $17$
Sign $-0.964 - 0.263i$
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  + (16.9 + 16.9i)2-s + (74.3 + 179. i)3-s + 64.3i·4-s + (−2.35e3 + 973. i)5-s + (−1.78e3 + 4.30e3i)6-s + (1.81e3 + 751. i)7-s + (7.59e3 − 7.59e3i)8-s + (−1.27e4 + 1.27e4i)9-s + (−5.64e4 − 2.33e4i)10-s + (6.97e3 − 1.68e4i)11-s + (−1.15e4 + 4.78e3i)12-s + 1.65e5i·13-s + (1.80e4 + 4.35e4i)14-s + (−3.49e5 − 3.49e5i)15-s + 2.90e5·16-s + (−6.54e4 + 3.38e5i)17-s + ⋯
L(s)  = 1  + (0.750 + 0.750i)2-s + (0.529 + 1.27i)3-s + 0.125i·4-s + (−1.68 + 0.696i)5-s + (−0.562 + 1.35i)6-s + (0.285 + 0.118i)7-s + (0.655 − 0.655i)8-s + (−0.648 + 0.648i)9-s + (−1.78 − 0.739i)10-s + (0.143 − 0.346i)11-s + (−0.160 + 0.0665i)12-s + 1.60i·13-s + (0.125 + 0.302i)14-s + (−1.78 − 1.78i)15-s + 1.10·16-s + (−0.189 + 0.981i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.263i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.964 - 0.263i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ -0.964 - 0.263i)\)

Particular Values

\(L(5)\) \(\approx\) \(0.287953 + 2.14363i\)
\(L(\frac12)\) \(\approx\) \(0.287953 + 2.14363i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (6.54e4 - 3.38e5i)T \)
good2 \( 1 + (-16.9 - 16.9i)T + 512iT^{2} \)
3 \( 1 + (-74.3 - 179. i)T + (-1.39e4 + 1.39e4i)T^{2} \)
5 \( 1 + (2.35e3 - 973. i)T + (1.38e6 - 1.38e6i)T^{2} \)
7 \( 1 + (-1.81e3 - 751. i)T + (2.85e7 + 2.85e7i)T^{2} \)
11 \( 1 + (-6.97e3 + 1.68e4i)T + (-1.66e9 - 1.66e9i)T^{2} \)
13 \( 1 - 1.65e5iT - 1.06e10T^{2} \)
19 \( 1 + (9.72e4 + 9.72e4i)T + 3.22e11iT^{2} \)
23 \( 1 + (2.66e5 - 6.42e5i)T + (-1.27e12 - 1.27e12i)T^{2} \)
29 \( 1 + (-3.90e6 + 1.61e6i)T + (1.02e13 - 1.02e13i)T^{2} \)
31 \( 1 + (-1.44e6 - 3.50e6i)T + (-1.86e13 + 1.86e13i)T^{2} \)
37 \( 1 + (-4.54e6 - 1.09e7i)T + (-9.18e13 + 9.18e13i)T^{2} \)
41 \( 1 + (9.08e6 + 3.76e6i)T + (2.31e14 + 2.31e14i)T^{2} \)
43 \( 1 + (-2.50e7 + 2.50e7i)T - 5.02e14iT^{2} \)
47 \( 1 + 5.08e7iT - 1.11e15T^{2} \)
53 \( 1 + (1.09e7 + 1.09e7i)T + 3.29e15iT^{2} \)
59 \( 1 + (6.57e7 - 6.57e7i)T - 8.66e15iT^{2} \)
61 \( 1 + (-1.88e8 - 7.81e7i)T + (8.26e15 + 8.26e15i)T^{2} \)
67 \( 1 + 5.35e7T + 2.72e16T^{2} \)
71 \( 1 + (5.11e7 + 1.23e8i)T + (-3.24e16 + 3.24e16i)T^{2} \)
73 \( 1 + (-4.04e7 + 1.67e7i)T + (4.16e16 - 4.16e16i)T^{2} \)
79 \( 1 + (5.24e7 - 1.26e8i)T + (-8.47e16 - 8.47e16i)T^{2} \)
83 \( 1 + (2.90e8 + 2.90e8i)T + 1.86e17iT^{2} \)
89 \( 1 - 2.89e8iT - 3.50e17T^{2} \)
97 \( 1 + (-2.61e8 + 1.08e8i)T + (5.37e17 - 5.37e17i)T^{2} \)
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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

−16.47428015261393061355685060853, −15.60407251209329335536819402896, −14.95475060654385426128675603168, −14.05783037491342855900326042171, −11.72480289657472913222075801096, −10.40163861459722009397813064815, −8.500775357410407481909227166811, −6.81494326405852293520630822376, −4.49787069465201684461842647350, −3.71190566556529794932326163624, 0.865662797836297744395558625475, 2.90523523835195394168116930723, 4.56040358040895556196564079798, 7.58359373353851393849781788077, 8.204007792835576401374598022231, 11.22753936854724217803514745590, 12.42375905284696191099501901275, 12.85460873432139384452542579637, 14.36914964146034846710376232214, 15.93057303684148063986171423221

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