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Properties

Label 2-17-17.13-c9-0-6
Degree $2$
Conductor $17$
Sign $0.342 + 0.939i$
Analytic cond. $8.75560$
Root an. cond. $2.95898$
Motivic weight $9$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 12.9i·2-s + (−71.7 − 71.7i)3-s + 345.·4-s + (1.46e3 + 1.46e3i)5-s + (−926. + 926. i)6-s + (1.52e3 − 1.52e3i)7-s − 1.10e4i·8-s − 9.39e3i·9-s + (1.89e4 − 1.89e4i)10-s + (9.29e3 − 9.29e3i)11-s + (−2.47e4 − 2.47e4i)12-s + 1.78e4·13-s + (−1.97e4 − 1.97e4i)14-s − 2.10e5i·15-s + 3.37e4·16-s + (1.82e5 − 2.92e5i)17-s + ⋯
L(s)  = 1  − 0.570i·2-s + (−0.511 − 0.511i)3-s + 0.674·4-s + (1.05 + 1.05i)5-s + (−0.291 + 0.291i)6-s + (0.240 − 0.240i)7-s − 0.955i·8-s − 0.477i·9-s + (0.600 − 0.600i)10-s + (0.191 − 0.191i)11-s + (−0.344 − 0.344i)12-s + 0.173·13-s + (−0.137 − 0.137i)14-s − 1.07i·15-s + 0.128·16-s + (0.529 − 0.848i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.342 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.342 + 0.939i$
Analytic conductor: \(8.75560\)
Root analytic conductor: \(2.95898\)
Motivic weight: \(9\)
Rational: no
Arithmetic: yes
Character: $\chi_{17} (13, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 17,\ (\ :9/2),\ 0.342 + 0.939i)\)

Particular Values

\(L(5)\) \(\approx\) \(1.69996 - 1.19001i\)
\(L(\frac12)\) \(\approx\) \(1.69996 - 1.19001i\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-1.82e5 + 2.92e5i)T \)
good2 \( 1 + 12.9iT - 512T^{2} \)
3 \( 1 + (71.7 + 71.7i)T + 1.96e4iT^{2} \)
5 \( 1 + (-1.46e3 - 1.46e3i)T + 1.95e6iT^{2} \)
7 \( 1 + (-1.52e3 + 1.52e3i)T - 4.03e7iT^{2} \)
11 \( 1 + (-9.29e3 + 9.29e3i)T - 2.35e9iT^{2} \)
13 \( 1 - 1.78e4T + 1.06e10T^{2} \)
19 \( 1 - 3.09e4iT - 3.22e11T^{2} \)
23 \( 1 + (-1.36e5 + 1.36e5i)T - 1.80e12iT^{2} \)
29 \( 1 + (-1.29e6 - 1.29e6i)T + 1.45e13iT^{2} \)
31 \( 1 + (-3.71e5 - 3.71e5i)T + 2.64e13iT^{2} \)
37 \( 1 + (-1.26e7 - 1.26e7i)T + 1.29e14iT^{2} \)
41 \( 1 + (1.08e7 - 1.08e7i)T - 3.27e14iT^{2} \)
43 \( 1 + 4.23e7iT - 5.02e14T^{2} \)
47 \( 1 + 5.91e7T + 1.11e15T^{2} \)
53 \( 1 - 8.32e7iT - 3.29e15T^{2} \)
59 \( 1 - 9.24e7iT - 8.66e15T^{2} \)
61 \( 1 + (7.20e7 - 7.20e7i)T - 1.16e16iT^{2} \)
67 \( 1 - 1.25e8T + 2.72e16T^{2} \)
71 \( 1 + (1.51e8 + 1.51e8i)T + 4.58e16iT^{2} \)
73 \( 1 + (7.28e7 + 7.28e7i)T + 5.88e16iT^{2} \)
79 \( 1 + (-6.78e6 + 6.78e6i)T - 1.19e17iT^{2} \)
83 \( 1 - 6.13e8iT - 1.86e17T^{2} \)
89 \( 1 + 5.89e8T + 3.50e17T^{2} \)
97 \( 1 + (-6.15e8 - 6.15e8i)T + 7.60e17iT^{2} \)
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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

−16.74736598948029348310969073375, −14.99252573471924468014764690270, −13.64159281683292228411492192662, −12.08515798982484078589839581929, −10.98758039302370425414920233111, −9.811001802497614747215437666983, −7.07028128187951445324820930015, −6.08521138392756148757704522475, −2.96318066753666989913155077893, −1.29457586623735194530267984470, 1.79706665202035675286517367591, 5.02500805590169822072763514791, 6.07655311719696572601605852935, 8.201806959955653414150307259782, 9.905534784910281193933240183624, 11.37401778445624889563146478802, 12.95496026056648069615956371229, 14.55710277313820958467558957827, 16.02979126538395808939225671983, 16.80261844100943077783702945791

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