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Properties

Label 2-17-17.16-c11-0-1
Degree $2$
Conductor $17$
Sign $-0.999 + 0.0231i$
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  + 37.8·2-s + 586. i·3-s − 616.·4-s − 6.25e3i·5-s + 2.22e4i·6-s + 6.34e4i·7-s − 1.00e5·8-s − 1.67e5·9-s − 2.36e5i·10-s − 5.18e5i·11-s − 3.61e5i·12-s − 2.42e6·13-s + 2.40e6i·14-s + 3.66e6·15-s − 2.55e6·16-s + (−5.85e6 + 1.35e5i)17-s + ⋯
L(s)  = 1  + 0.835·2-s + 1.39i·3-s − 0.301·4-s − 0.894i·5-s + 1.16i·6-s + 1.42i·7-s − 1.08·8-s − 0.944·9-s − 0.748i·10-s − 0.970i·11-s − 0.419i·12-s − 1.80·13-s + 1.19i·14-s + 1.24·15-s − 0.608·16-s + (−0.999 + 0.0231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0231i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0231i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.999 + 0.0231i$
Analytic conductor: \(13.0618\)
Root analytic conductor: \(3.61411\)
Motivic weight: \(11\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :11/2),\ -0.999 + 0.0231i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.0120733 - 1.04476i\)
\(L(\frac12)\) \(\approx\) \(0.0120733 - 1.04476i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (5.85e6 - 1.35e5i)T \)
good2 \( 1 - 37.8T + 2.04e3T^{2} \)
3 \( 1 - 586. iT - 1.77e5T^{2} \)
5 \( 1 + 6.25e3iT - 4.88e7T^{2} \)
7 \( 1 - 6.34e4iT - 1.97e9T^{2} \)
11 \( 1 + 5.18e5iT - 2.85e11T^{2} \)
13 \( 1 + 2.42e6T + 1.79e12T^{2} \)
19 \( 1 - 1.20e7T + 1.16e14T^{2} \)
23 \( 1 - 3.49e7iT - 9.52e14T^{2} \)
29 \( 1 - 1.48e8iT - 1.22e16T^{2} \)
31 \( 1 - 5.98e7iT - 2.54e16T^{2} \)
37 \( 1 + 2.67e8iT - 1.77e17T^{2} \)
41 \( 1 - 7.79e8iT - 5.50e17T^{2} \)
43 \( 1 - 4.04e8T + 9.29e17T^{2} \)
47 \( 1 + 2.03e9T + 2.47e18T^{2} \)
53 \( 1 - 1.66e8T + 9.26e18T^{2} \)
59 \( 1 - 4.86e9T + 3.01e19T^{2} \)
61 \( 1 + 3.94e9iT - 4.35e19T^{2} \)
67 \( 1 + 1.51e10T + 1.22e20T^{2} \)
71 \( 1 - 5.71e9iT - 2.31e20T^{2} \)
73 \( 1 + 1.52e10iT - 3.13e20T^{2} \)
79 \( 1 - 2.11e10iT - 7.47e20T^{2} \)
83 \( 1 - 9.52e9T + 1.28e21T^{2} \)
89 \( 1 + 2.35e10T + 2.77e21T^{2} \)
97 \( 1 + 3.53e10iT - 7.15e21T^{2} \)
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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

−16.39810580719659690830937964397, −15.43290370228186430970519315592, −14.44449248589361884016353835201, −12.80651684713013264380779633475, −11.64927034381965985630005127289, −9.509930496648060865354231573843, −8.845546921932533461348558618246, −5.41054270202741401959929923076, −4.83742570245685839649436358410, −3.10633900271033485046176514521, 0.32759888300012882016133754295, 2.50488353892302850614453125620, 4.50461378352734666998382354987, 6.72203931775853142205344487398, 7.48547466930752452389295330716, 10.00106454927284606911619383687, 11.90970041137031309390441257794, 13.03585205585250952283036756602, 13.96921767020133547600570821709, 14.86547091613402004152466313678

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