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Properties

Label 2-17-17.13-c11-0-14
Degree $2$
Conductor $17$
Sign $0.123 - 0.992i$
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  − 44.2i·2-s + (−276. − 276. i)3-s + 92.6·4-s + (−5.75e3 − 5.75e3i)5-s + (−1.22e4 + 1.22e4i)6-s + (−2.32e4 + 2.32e4i)7-s − 9.46e4i·8-s − 2.38e4i·9-s + (−2.54e5 + 2.54e5i)10-s + (−5.02e5 + 5.02e5i)11-s + (−2.56e4 − 2.56e4i)12-s + 1.89e6·13-s + (1.03e6 + 1.03e6i)14-s + 3.18e6i·15-s − 3.99e6·16-s + (5.02e6 − 3.00e6i)17-s + ⋯
L(s)  = 1  − 0.977i·2-s + (−0.657 − 0.657i)3-s + 0.0452·4-s + (−0.823 − 0.823i)5-s + (−0.642 + 0.642i)6-s + (−0.523 + 0.523i)7-s − 1.02i·8-s − 0.134i·9-s + (−0.804 + 0.804i)10-s + (−0.941 + 0.941i)11-s + (−0.0297 − 0.0297i)12-s + 1.41·13-s + (0.511 + 0.511i)14-s + 1.08i·15-s − 0.952·16-s + (0.858 − 0.513i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $0.123 - 0.992i$
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: \((2,\ 17,\ (\ :11/2),\ 0.123 - 0.992i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.240213 + 0.212091i\)
\(L(\frac12)\) \(\approx\) \(0.240213 + 0.212091i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-5.02e6 + 3.00e6i)T \)
good2 \( 1 + 44.2iT - 2.04e3T^{2} \)
3 \( 1 + (276. + 276. i)T + 1.77e5iT^{2} \)
5 \( 1 + (5.75e3 + 5.75e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (2.32e4 - 2.32e4i)T - 1.97e9iT^{2} \)
11 \( 1 + (5.02e5 - 5.02e5i)T - 2.85e11iT^{2} \)
13 \( 1 - 1.89e6T + 1.79e12T^{2} \)
19 \( 1 - 1.66e7iT - 1.16e14T^{2} \)
23 \( 1 + (1.38e7 - 1.38e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (1.26e8 + 1.26e8i)T + 1.22e16iT^{2} \)
31 \( 1 + (7.43e7 + 7.43e7i)T + 2.54e16iT^{2} \)
37 \( 1 + (1.64e8 + 1.64e8i)T + 1.77e17iT^{2} \)
41 \( 1 + (7.59e7 - 7.59e7i)T - 5.50e17iT^{2} \)
43 \( 1 - 2.71e8iT - 9.29e17T^{2} \)
47 \( 1 + 2.59e9T + 2.47e18T^{2} \)
53 \( 1 - 2.36e9iT - 9.26e18T^{2} \)
59 \( 1 + 6.06e9iT - 3.01e19T^{2} \)
61 \( 1 + (-4.43e8 + 4.43e8i)T - 4.35e19iT^{2} \)
67 \( 1 + 2.01e9T + 1.22e20T^{2} \)
71 \( 1 + (-1.22e9 - 1.22e9i)T + 2.31e20iT^{2} \)
73 \( 1 + (-1.25e10 - 1.25e10i)T + 3.13e20iT^{2} \)
79 \( 1 + (-9.09e9 + 9.09e9i)T - 7.47e20iT^{2} \)
83 \( 1 + 6.16e10iT - 1.28e21T^{2} \)
89 \( 1 + 4.40e10T + 2.77e21T^{2} \)
97 \( 1 + (9.82e10 + 9.82e10i)T + 7.15e21iT^{2} \)
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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

−15.52722274029186451167259902091, −12.93808343938873580657265004008, −12.32512694190798104757409823690, −11.42243898077847131097417061400, −9.738467758732107324704519312563, −7.74361898007654147141697632401, −5.92736865860747388534116818103, −3.67181847709404362816828165640, −1.56808748563605576288151802489, −0.14691730373847010087789528276, 3.40579191066354152161993074088, 5.45623946629946547062020025522, 6.83862132376720690172562403741, 8.204398027488380480883324582410, 10.72107067778482406369550395887, 11.17847139747859695195831845449, 13.52987502649037736130308240562, 15.10528341369640826148593898821, 16.07371903281415701990149935858, 16.55234556718957392441080568244

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