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Properties

Label 2-17-17.13-c11-0-2
Degree $2$
Conductor $17$
Sign $0.130 - 0.991i$
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  + 66.0i·2-s + (−572. − 572. i)3-s − 2.30e3·4-s + (−7.46e3 − 7.46e3i)5-s + (3.78e4 − 3.78e4i)6-s + (−2.46e4 + 2.46e4i)7-s − 1.72e4i·8-s + 4.79e5i·9-s + (4.92e5 − 4.92e5i)10-s + (3.98e5 − 3.98e5i)11-s + (1.32e6 + 1.32e6i)12-s − 2.26e5·13-s + (−1.62e6 − 1.62e6i)14-s + 8.54e6i·15-s − 3.59e6·16-s + (5.04e6 − 2.97e6i)17-s + ⋯
L(s)  = 1  + 1.45i·2-s + (−1.36 − 1.36i)3-s − 1.12·4-s + (−1.06 − 1.06i)5-s + (1.98 − 1.98i)6-s + (−0.554 + 0.554i)7-s − 0.185i·8-s + 2.70i·9-s + (1.55 − 1.55i)10-s + (0.746 − 0.746i)11-s + (1.53 + 1.53i)12-s − 0.168·13-s + (−0.808 − 0.808i)14-s + 2.90i·15-s − 0.856·16-s + (0.861 − 0.507i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.130 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.130 - 0.991i$
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: \((2,\ 17,\ (\ :11/2),\ 0.130 - 0.991i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.415078 + 0.364156i\)
\(L(\frac12)\) \(\approx\) \(0.415078 + 0.364156i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-5.04e6 + 2.97e6i)T \)
good2 \( 1 - 66.0iT - 2.04e3T^{2} \)
3 \( 1 + (572. + 572. i)T + 1.77e5iT^{2} \)
5 \( 1 + (7.46e3 + 7.46e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (2.46e4 - 2.46e4i)T - 1.97e9iT^{2} \)
11 \( 1 + (-3.98e5 + 3.98e5i)T - 2.85e11iT^{2} \)
13 \( 1 + 2.26e5T + 1.79e12T^{2} \)
19 \( 1 + 2.30e6iT - 1.16e14T^{2} \)
23 \( 1 + (1.59e7 - 1.59e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (-2.98e7 - 2.98e7i)T + 1.22e16iT^{2} \)
31 \( 1 + (-1.34e8 - 1.34e8i)T + 2.54e16iT^{2} \)
37 \( 1 + (8.02e7 + 8.02e7i)T + 1.77e17iT^{2} \)
41 \( 1 + (5.65e8 - 5.65e8i)T - 5.50e17iT^{2} \)
43 \( 1 + 6.61e8iT - 9.29e17T^{2} \)
47 \( 1 - 1.01e9T + 2.47e18T^{2} \)
53 \( 1 - 1.72e9iT - 9.26e18T^{2} \)
59 \( 1 - 5.73e9iT - 3.01e19T^{2} \)
61 \( 1 + (-5.29e9 + 5.29e9i)T - 4.35e19iT^{2} \)
67 \( 1 + 1.57e9T + 1.22e20T^{2} \)
71 \( 1 + (-1.57e10 - 1.57e10i)T + 2.31e20iT^{2} \)
73 \( 1 + (1.72e10 + 1.72e10i)T + 3.13e20iT^{2} \)
79 \( 1 + (-8.68e9 + 8.68e9i)T - 7.47e20iT^{2} \)
83 \( 1 - 3.00e10iT - 1.28e21T^{2} \)
89 \( 1 - 5.49e10T + 2.77e21T^{2} \)
97 \( 1 + (5.33e9 + 5.33e9i)T + 7.15e21iT^{2} \)
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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

−16.48772913568617848558005323042, −15.85332534030567627871772444433, −13.77347907819282311088810693772, −12.35413731130952658550597698603, −11.64043894487322681886272042008, −8.532731139724735746178533156950, −7.39896718975126446066018310317, −6.16622799377013179800426509040, −5.06498074897596363650113728180, −0.831391922223681469686495374699, 0.43152234548069804291146086958, 3.49231828049808682402463115181, 4.23893872127467862241664311141, 6.64069150821223460599082645989, 9.861383706228176017013110845694, 10.44212794013907405996288727743, 11.55978442869401606417821796163, 12.22940382745542058265256825481, 14.79773244290638703474205322465, 15.95144583588007340947627194727

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