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Properties

Label 2-17-17.13-c5-0-2
Degree $2$
Conductor $17$
Sign $0.860 - 0.509i$
Analytic cond. $2.72652$
Root an. cond. $1.65121$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.52i·2-s + (−1.82 − 1.82i)3-s + 29.6·4-s + (52.9 + 52.9i)5-s + (2.76 − 2.76i)6-s + (−25.6 + 25.6i)7-s + 93.8i·8-s − 236. i·9-s + (−80.5 + 80.5i)10-s + (172. − 172. i)11-s + (−54.0 − 54.0i)12-s − 949.·13-s + (−38.9 − 38.9i)14-s − 192. i·15-s + 807.·16-s + (−1.10e3 − 434. i)17-s + ⋯
L(s)  = 1  + 0.268i·2-s + (−0.116 − 0.116i)3-s + 0.927·4-s + (0.947 + 0.947i)5-s + (0.0314 − 0.0314i)6-s + (−0.197 + 0.197i)7-s + 0.518i·8-s − 0.972i·9-s + (−0.254 + 0.254i)10-s + (0.430 − 0.430i)11-s + (−0.108 − 0.108i)12-s − 1.55·13-s + (−0.0531 − 0.0531i)14-s − 0.221i·15-s + 0.788·16-s + (−0.931 − 0.364i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.860 - 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.860 - 0.509i$
Analytic conductor: \(2.72652\)
Root analytic conductor: \(1.65121\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :5/2),\ 0.860 - 0.509i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.57543 + 0.431717i\)
\(L(\frac12)\) \(\approx\) \(1.57543 + 0.431717i\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (1.10e3 + 434. i)T \)
good2 \( 1 - 1.52iT - 32T^{2} \)
3 \( 1 + (1.82 + 1.82i)T + 243iT^{2} \)
5 \( 1 + (-52.9 - 52.9i)T + 3.12e3iT^{2} \)
7 \( 1 + (25.6 - 25.6i)T - 1.68e4iT^{2} \)
11 \( 1 + (-172. + 172. i)T - 1.61e5iT^{2} \)
13 \( 1 + 949.T + 3.71e5T^{2} \)
19 \( 1 - 933. iT - 2.47e6T^{2} \)
23 \( 1 + (-2.22e3 + 2.22e3i)T - 6.43e6iT^{2} \)
29 \( 1 + (1.56e3 + 1.56e3i)T + 2.05e7iT^{2} \)
31 \( 1 + (2.92e3 + 2.92e3i)T + 2.86e7iT^{2} \)
37 \( 1 + (3.49e3 + 3.49e3i)T + 6.93e7iT^{2} \)
41 \( 1 + (-3.76e3 + 3.76e3i)T - 1.15e8iT^{2} \)
43 \( 1 - 2.07e4iT - 1.47e8T^{2} \)
47 \( 1 - 1.67e4T + 2.29e8T^{2} \)
53 \( 1 - 185. iT - 4.18e8T^{2} \)
59 \( 1 + 2.19e4iT - 7.14e8T^{2} \)
61 \( 1 + (2.53e4 - 2.53e4i)T - 8.44e8iT^{2} \)
67 \( 1 + 2.73e4T + 1.35e9T^{2} \)
71 \( 1 + (-5.40e4 - 5.40e4i)T + 1.80e9iT^{2} \)
73 \( 1 + (-1.34e4 - 1.34e4i)T + 2.07e9iT^{2} \)
79 \( 1 + (-6.69e4 + 6.69e4i)T - 3.07e9iT^{2} \)
83 \( 1 - 8.07e4iT - 3.93e9T^{2} \)
89 \( 1 - 3.51e4T + 5.58e9T^{2} \)
97 \( 1 + (7.17e4 + 7.17e4i)T + 8.58e9iT^{2} \)
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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

−17.79853041773449828942913351011, −16.79189300122026141942165266026, −15.13182955510690944852331795078, −14.33728697133540721310598742769, −12.39389978846878278179039935473, −10.99162952914209634884067619196, −9.513630737051363211069689222673, −7.06052774588270874981336273571, −6.06674347840798688497158214040, −2.57753021960034991276718147964, 2.00534215702332649451104958284, 5.13850866704962534746546682772, 7.12859510029448137221473594118, 9.353769341663058279346362339472, 10.68609228211202934902140939823, 12.29399186694529976799609065214, 13.46567067554217723904292928099, 15.24677297483785273837086332085, 16.71659226844191545357932245973, 17.29421862527920096416330606017

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