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Properties

Label 2-17-17.13-c11-0-1
Degree $2$
Conductor $17$
Sign $0.999 - 0.0355i$
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  − 66.7i·2-s + (−522. − 522. i)3-s − 2.40e3·4-s + (7.52e3 + 7.52e3i)5-s + (−3.49e4 + 3.49e4i)6-s + (−2.30e4 + 2.30e4i)7-s + 2.39e4i·8-s + 3.69e5i·9-s + (5.02e5 − 5.02e5i)10-s + (1.28e5 − 1.28e5i)11-s + (1.25e6 + 1.25e6i)12-s − 2.00e6·13-s + (1.53e6 + 1.53e6i)14-s − 7.86e6i·15-s − 3.33e6·16-s + (3.76e6 + 4.48e6i)17-s + ⋯
L(s)  = 1  − 1.47i·2-s + (−1.24 − 1.24i)3-s − 1.17·4-s + (1.07 + 1.07i)5-s + (−1.83 + 1.83i)6-s + (−0.518 + 0.518i)7-s + 0.258i·8-s + 2.08i·9-s + (1.58 − 1.58i)10-s + (0.239 − 0.239i)11-s + (1.46 + 1.46i)12-s − 1.49·13-s + (0.764 + 0.764i)14-s − 2.67i·15-s − 0.793·16-s + (0.643 + 0.765i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0355i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.999 - 0.0355i$
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: \((2,\ 17,\ (\ :11/2),\ 0.999 - 0.0355i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.493958 + 0.00877912i\)
\(L(\frac12)\) \(\approx\) \(0.493958 + 0.00877912i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.76e6 - 4.48e6i)T \)
good2 \( 1 + 66.7iT - 2.04e3T^{2} \)
3 \( 1 + (522. + 522. i)T + 1.77e5iT^{2} \)
5 \( 1 + (-7.52e3 - 7.52e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (2.30e4 - 2.30e4i)T - 1.97e9iT^{2} \)
11 \( 1 + (-1.28e5 + 1.28e5i)T - 2.85e11iT^{2} \)
13 \( 1 + 2.00e6T + 1.79e12T^{2} \)
19 \( 1 - 6.91e6iT - 1.16e14T^{2} \)
23 \( 1 + (-3.18e6 + 3.18e6i)T - 9.52e14iT^{2} \)
29 \( 1 + (-1.17e8 - 1.17e8i)T + 1.22e16iT^{2} \)
31 \( 1 + (1.60e8 + 1.60e8i)T + 2.54e16iT^{2} \)
37 \( 1 + (3.63e8 + 3.63e8i)T + 1.77e17iT^{2} \)
41 \( 1 + (4.12e8 - 4.12e8i)T - 5.50e17iT^{2} \)
43 \( 1 - 7.21e8iT - 9.29e17T^{2} \)
47 \( 1 - 7.23e8T + 2.47e18T^{2} \)
53 \( 1 + 2.18e8iT - 9.26e18T^{2} \)
59 \( 1 + 4.16e8iT - 3.01e19T^{2} \)
61 \( 1 + (6.64e9 - 6.64e9i)T - 4.35e19iT^{2} \)
67 \( 1 - 3.24e9T + 1.22e20T^{2} \)
71 \( 1 + (-8.45e9 - 8.45e9i)T + 2.31e20iT^{2} \)
73 \( 1 + (-5.17e9 - 5.17e9i)T + 3.13e20iT^{2} \)
79 \( 1 + (3.77e10 - 3.77e10i)T - 7.47e20iT^{2} \)
83 \( 1 + 2.98e10iT - 1.28e21T^{2} \)
89 \( 1 + 5.62e10T + 2.77e21T^{2} \)
97 \( 1 + (4.57e10 + 4.57e10i)T + 7.15e21iT^{2} \)
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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

−16.89733470769450239698369739507, −14.25556689834874297005880491273, −12.82293139260467110732736767414, −12.15039933076479053567785152610, −10.85222426381332289988065255378, −9.871058759282948414239415022394, −6.94215549860576416579362281992, −5.73017854329733813394682459208, −2.67720109585779874592895477874, −1.54976830392447559665378209296, 0.24267014985101060674009211355, 4.76754574092279306082083035751, 5.39860328000917339305715262750, 6.81648536709903137358512158010, 9.220834313429699231025785476707, 10.08442780008705921680180569161, 12.16241519910616732863821445237, 13.91348579772698471065276872702, 15.41006680417985033056523742745, 16.46177236586994289284997240808

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