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Properties

Label 2-17-17.13-c11-0-8
Degree $2$
Conductor $17$
Sign $0.902 - 0.430i$
Analytic cond. $13.0618$
Root an. cond. $3.61411$
Motivic weight $11$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 48.9i·2-s + (−20.4 − 20.4i)3-s − 347.·4-s + (−2.62e3 − 2.62e3i)5-s + (1.00e3 − 1.00e3i)6-s + (4.67e4 − 4.67e4i)7-s + 8.32e4i·8-s − 1.76e5i·9-s + (1.28e5 − 1.28e5i)10-s + (1.98e5 − 1.98e5i)11-s + (7.10e3 + 7.10e3i)12-s + 5.35e5·13-s + (2.28e6 + 2.28e6i)14-s + 1.07e5i·15-s − 4.78e6·16-s + (5.23e6 + 2.61e6i)17-s + ⋯
L(s)  = 1  + 1.08i·2-s + (−0.0486 − 0.0486i)3-s − 0.169·4-s + (−0.376 − 0.376i)5-s + (0.0525 − 0.0525i)6-s + (1.05 − 1.05i)7-s + 0.898i·8-s − 0.995i·9-s + (0.406 − 0.406i)10-s + (0.371 − 0.371i)11-s + (0.00824 + 0.00824i)12-s + 0.399·13-s + (1.13 + 1.13i)14-s + 0.0365i·15-s − 1.14·16-s + (0.894 + 0.446i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.902 - 0.430i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.902 - 0.430i$
Analytic conductor: \(13.0618\)
Root analytic conductor: \(3.61411\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :11/2),\ 0.902 - 0.430i)\)

Particular Values

\(L(6)\) \(\approx\) \(2.02126 + 0.457431i\)
\(L(\frac12)\) \(\approx\) \(2.02126 + 0.457431i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-5.23e6 - 2.61e6i)T \)
good2 \( 1 - 48.9iT - 2.04e3T^{2} \)
3 \( 1 + (20.4 + 20.4i)T + 1.77e5iT^{2} \)
5 \( 1 + (2.62e3 + 2.62e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (-4.67e4 + 4.67e4i)T - 1.97e9iT^{2} \)
11 \( 1 + (-1.98e5 + 1.98e5i)T - 2.85e11iT^{2} \)
13 \( 1 - 5.35e5T + 1.79e12T^{2} \)
19 \( 1 - 1.86e7iT - 1.16e14T^{2} \)
23 \( 1 + (-3.97e7 + 3.97e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (-2.42e7 - 2.42e7i)T + 1.22e16iT^{2} \)
31 \( 1 + (1.83e8 + 1.83e8i)T + 2.54e16iT^{2} \)
37 \( 1 + (-1.97e8 - 1.97e8i)T + 1.77e17iT^{2} \)
41 \( 1 + (-6.99e8 + 6.99e8i)T - 5.50e17iT^{2} \)
43 \( 1 + 9.40e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.89e8T + 2.47e18T^{2} \)
53 \( 1 - 4.79e9iT - 9.26e18T^{2} \)
59 \( 1 - 1.75e9iT - 3.01e19T^{2} \)
61 \( 1 + (3.37e9 - 3.37e9i)T - 4.35e19iT^{2} \)
67 \( 1 + 4.86e9T + 1.22e20T^{2} \)
71 \( 1 + (-3.38e9 - 3.38e9i)T + 2.31e20iT^{2} \)
73 \( 1 + (1.02e10 + 1.02e10i)T + 3.13e20iT^{2} \)
79 \( 1 + (2.65e10 - 2.65e10i)T - 7.47e20iT^{2} \)
83 \( 1 - 2.97e10iT - 1.28e21T^{2} \)
89 \( 1 - 5.39e10T + 2.77e21T^{2} \)
97 \( 1 + (3.77e8 + 3.77e8i)T + 7.15e21iT^{2} \)
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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

−16.51741572006290335794652457958, −14.90101306589794956609800294570, −14.20472100699388815165510920676, −12.20409903968982631605999759023, −10.79016168220910403605631310594, −8.559333889734925316172078697398, −7.45975085770247081550834782349, −5.94091545126665549939209710924, −4.09524895355629203585139916160, −1.05510334043899702221215097874, 1.52356791311610455613515219549, 2.94508107961884752509851564588, 5.02024319673205432036289388838, 7.38869443017116460005663390878, 9.226595851120772535934470828803, 11.08232727000904821731437566240, 11.51309639071194327452911418906, 13.06278129986198136533425001096, 14.79021104447183306804154167496, 15.95530261220310229149330679595

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