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Properties

Label 2-17e2-17.15-c1-0-13
Degree $2$
Conductor $289$
Sign $-0.722 - 0.691i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.921 − 0.921i)2-s + (−0.881 − 2.12i)3-s − 0.302i·4-s + (1.20 − 0.498i)5-s + (−1.14 + 2.77i)6-s + (−3.05 − 1.26i)7-s + (−2.12 + 2.12i)8-s + (−1.62 + 1.62i)9-s + (−1.56 − 0.649i)10-s + (1.14 − 2.77i)11-s + (−0.644 + 0.266i)12-s − 0.302i·13-s + (1.64 + 3.97i)14-s + (−2.12 − 2.12i)15-s + 3.30·16-s + ⋯
L(s)  = 1  + (−0.651 − 0.651i)2-s + (−0.508 − 1.22i)3-s − 0.151i·4-s + (0.538 − 0.222i)5-s + (−0.468 + 1.13i)6-s + (−1.15 − 0.477i)7-s + (−0.749 + 0.749i)8-s + (−0.542 + 0.542i)9-s + (−0.495 − 0.205i)10-s + (0.346 − 0.835i)11-s + (−0.185 + 0.0770i)12-s − 0.0839i·13-s + (0.440 + 1.06i)14-s + (−0.547 − 0.547i)15-s + 0.825·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.722 - 0.691i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (134, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.206969 + 0.515415i\)
\(L(\frac12)\) \(\approx\) \(0.206969 + 0.515415i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + (0.921 + 0.921i)T + 2iT^{2} \)
3 \( 1 + (0.881 + 2.12i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-1.20 + 0.498i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (3.05 + 1.26i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1.14 + 2.77i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 0.302iT - 13T^{2} \)
19 \( 1 + (-3.47 - 3.47i)T + 19iT^{2} \)
23 \( 1 + (-0.498 + 1.20i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (0.839 - 0.347i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (1.37 + 3.33i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-2.52 - 6.10i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (5.54 + 2.29i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (-6.79 + 6.79i)T - 43iT^{2} \)
47 \( 1 + 3iT - 47T^{2} \)
53 \( 1 + (9.12 + 9.12i)T + 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (-11.8 - 4.90i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 5.39T + 67T^{2} \)
71 \( 1 + (4.29 + 10.3i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (7.02 - 2.91i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.22 - 2.96i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (10.9 + 10.9i)T + 83iT^{2} \)
89 \( 1 + 11.2iT - 89T^{2} \)
97 \( 1 + (0.0846 - 0.0350i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.33931121039279507191057231085, −10.16815577294568501264159906697, −9.580050955118657764849864942571, −8.488820737799944270752248873710, −7.22868136754547387966365056416, −6.19920409805482589018526939332, −5.62288271725660367842752301511, −3.35157256567893218599627363289, −1.75726175165870041980815966319, −0.54383058525805185336984178200, 2.96069487437256191303174288811, 4.22056001321089159466662989413, 5.60380888186862033736980202879, 6.49601404204613095399954242425, 7.45545879430068123288293471732, 8.996846691541762782705768461361, 9.578984488516269651409714779626, 10.01461705256966388251462331273, 11.27411260174023155064344672792, 12.32384110952827013708165737643

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