↑

Properties

Label 2-17-17.13-c11-0-12
Degree $2$
Conductor $17$
Sign $-0.901 + 0.431i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 49.5i·2-s + (136. + 136. i)3-s − 403.·4-s + (−1.14e3 − 1.14e3i)5-s + (6.74e3 − 6.74e3i)6-s + (−1.40e4 + 1.40e4i)7-s − 8.14e4i·8-s − 1.40e5i·9-s + (−5.64e4 + 5.64e4i)10-s + (6.23e5 − 6.23e5i)11-s + (−5.50e4 − 5.50e4i)12-s − 1.99e6·13-s + (6.95e5 + 6.95e5i)14-s − 3.10e5i·15-s − 4.85e6·16-s + (−5.24e6 − 2.60e6i)17-s + ⋯
L(s)  = 1  − 1.09i·2-s + (0.323 + 0.323i)3-s − 0.197·4-s + (−0.163 − 0.163i)5-s + (0.353 − 0.353i)6-s + (−0.315 + 0.315i)7-s − 0.878i·8-s − 0.790i·9-s + (−0.178 + 0.178i)10-s + (1.16 − 1.16i)11-s + (−0.0638 − 0.0638i)12-s − 1.49·13-s + (0.345 + 0.345i)14-s − 0.105i·15-s − 1.15·16-s + (−0.895 − 0.445i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & (-0.901 + 0.431i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(17\)
Sign: $-0.901 + 0.431i$
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: \((2,\ 17,\ (\ :11/2),\ -0.901 + 0.431i)\)

Particular Values

\(L(6)\) \(\approx\) \(0.368976 - 1.62499i\)
\(L(\frac12)\) \(\approx\) \(0.368976 - 1.62499i\)
\(L(\frac{13}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (5.24e6 + 2.60e6i)T \)
good2 \( 1 + 49.5iT - 2.04e3T^{2} \)
3 \( 1 + (-136. - 136. i)T + 1.77e5iT^{2} \)
5 \( 1 + (1.14e3 + 1.14e3i)T + 4.88e7iT^{2} \)
7 \( 1 + (1.40e4 - 1.40e4i)T - 1.97e9iT^{2} \)
11 \( 1 + (-6.23e5 + 6.23e5i)T - 2.85e11iT^{2} \)
13 \( 1 + 1.99e6T + 1.79e12T^{2} \)
19 \( 1 - 1.15e7iT - 1.16e14T^{2} \)
23 \( 1 + (-1.65e7 + 1.65e7i)T - 9.52e14iT^{2} \)
29 \( 1 + (7.07e7 + 7.07e7i)T + 1.22e16iT^{2} \)
31 \( 1 + (-1.31e8 - 1.31e8i)T + 2.54e16iT^{2} \)
37 \( 1 + (-3.23e8 - 3.23e8i)T + 1.77e17iT^{2} \)
41 \( 1 + (-3.12e7 + 3.12e7i)T - 5.50e17iT^{2} \)
43 \( 1 + 1.54e9iT - 9.29e17T^{2} \)
47 \( 1 - 1.06e9T + 2.47e18T^{2} \)
53 \( 1 + 6.43e8iT - 9.26e18T^{2} \)
59 \( 1 - 6.17e9iT - 3.01e19T^{2} \)
61 \( 1 + (-3.25e9 + 3.25e9i)T - 4.35e19iT^{2} \)
67 \( 1 - 1.10e10T + 1.22e20T^{2} \)
71 \( 1 + (-1.82e10 - 1.82e10i)T + 2.31e20iT^{2} \)
73 \( 1 + (3.89e9 + 3.89e9i)T + 3.13e20iT^{2} \)
79 \( 1 + (1.29e10 - 1.29e10i)T - 7.47e20iT^{2} \)
83 \( 1 - 9.53e9iT - 1.28e21T^{2} \)
89 \( 1 - 2.08e10T + 2.77e21T^{2} \)
97 \( 1 + (6.90e10 + 6.90e10i)T + 7.15e21iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.57751137076560411479587850753, −14.20542184955766830769724925279, −12.43063485375047999960520910516, −11.63789382814639155013230146188, −10.00224033380335639401256955952, −8.903441000199494218960374466661, −6.54591950224024465469108881501, −4.01926364417334497682049730038, −2.66451263673678485623303004548, −0.66378857581867109653245331631, 2.20300282100523009417569478004, 4.77340647779446031247282009802, 6.82912475503887510145627972524, 7.54469224821860884910719051695, 9.341046314955617036500537069752, 11.30872814434694819752988379092, 13.04080326599018622574021862753, 14.50842539068649901949960623849, 15.29460702329118969214483264185, 16.83465721266852957469948058792

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