Properties

Label 2-17e2-289.128-c1-0-11
Degree $2$
Conductor $289$
Sign $-0.845 + 0.534i$
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  + (−2.71 − 0.379i)2-s + (0.249 − 0.0899i)3-s + (5.32 + 1.51i)4-s + (−2.08 + 2.18i)5-s + (−0.711 + 0.150i)6-s + (−0.129 + 0.0400i)7-s + (−8.87 − 3.91i)8-s + (−2.25 + 1.87i)9-s + (6.49 − 5.14i)10-s + (0.0853 + 0.735i)11-s + (1.46 − 0.101i)12-s + (−2.02 − 5.23i)13-s + (0.366 − 0.0597i)14-s + (−0.323 + 0.731i)15-s + (13.2 + 8.19i)16-s + (3.41 − 2.31i)17-s + ⋯
L(s)  = 1  + (−1.92 − 0.268i)2-s + (0.143 − 0.0519i)3-s + (2.66 + 0.757i)4-s + (−0.932 + 0.976i)5-s + (−0.290 + 0.0612i)6-s + (−0.0488 + 0.0151i)7-s + (−3.13 − 1.38i)8-s + (−0.751 + 0.623i)9-s + (2.05 − 1.62i)10-s + (0.0257 + 0.221i)11-s + (0.422 − 0.0293i)12-s + (−0.562 − 1.45i)13-s + (0.0979 − 0.0159i)14-s + (−0.0834 + 0.188i)15-s + (3.30 + 2.04i)16-s + (0.827 − 0.561i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 + 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.845 + 0.534i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (128, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.845 + 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0240505 - 0.0830803i\)
\(L(\frac12)\) \(\approx\) \(0.0240505 - 0.0830803i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 + (-3.41 + 2.31i)T \)
good2 \( 1 + (2.71 + 0.379i)T + (1.92 + 0.547i)T^{2} \)
3 \( 1 + (-0.249 + 0.0899i)T + (2.30 - 1.91i)T^{2} \)
5 \( 1 + (2.08 - 2.18i)T + (-0.230 - 4.99i)T^{2} \)
7 \( 1 + (0.129 - 0.0400i)T + (5.77 - 3.95i)T^{2} \)
11 \( 1 + (-0.0853 - 0.735i)T + (-10.7 + 2.51i)T^{2} \)
13 \( 1 + (2.02 + 5.23i)T + (-9.60 + 8.75i)T^{2} \)
19 \( 1 + (0.150 + 1.08i)T + (-18.2 + 5.19i)T^{2} \)
23 \( 1 + (1.35 + 4.37i)T + (-18.9 + 12.9i)T^{2} \)
29 \( 1 + (5.38 + 4.26i)T + (6.63 + 28.2i)T^{2} \)
31 \( 1 + (8.60 + 0.198i)T + (30.9 + 1.43i)T^{2} \)
37 \( 1 + (-4.66 + 4.06i)T + (5.11 - 36.6i)T^{2} \)
41 \( 1 + (3.63 + 1.70i)T + (26.1 + 31.5i)T^{2} \)
43 \( 1 + (-0.0937 - 0.0220i)T + (38.4 + 19.1i)T^{2} \)
47 \( 1 + (5.29 - 0.490i)T + (46.1 - 8.63i)T^{2} \)
53 \( 1 + (-3.42 - 4.11i)T + (-9.73 + 52.0i)T^{2} \)
59 \( 1 + (-6.04 - 4.14i)T + (21.3 + 55.0i)T^{2} \)
61 \( 1 + (0.0669 + 0.317i)T + (-55.8 + 24.6i)T^{2} \)
67 \( 1 + (7.57 - 10.0i)T + (-18.3 - 64.4i)T^{2} \)
71 \( 1 + (-6.70 + 12.7i)T + (-40.1 - 58.5i)T^{2} \)
73 \( 1 + (3.37 - 2.42i)T + (23.1 - 69.2i)T^{2} \)
79 \( 1 + (-0.903 + 3.47i)T + (-69.0 - 38.4i)T^{2} \)
83 \( 1 + (14.5 - 0.672i)T + (82.6 - 7.65i)T^{2} \)
89 \( 1 + (0.166 - 0.430i)T + (-65.7 - 59.9i)T^{2} \)
97 \( 1 + (8.21 + 4.32i)T + (54.8 + 80.0i)T^{2} \)
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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

−11.09241727116649619197805088200, −10.48039232028440428985446776033, −9.602287833777195673743130590391, −8.433852653796863309354131502614, −7.63129302011210313142709272733, −7.28078106072979022212427671132, −5.79897703058492824650408907328, −3.30069774193985795535476760658, −2.45565483106485841479492133395, −0.11247997722320052674372378720, 1.61258602013516481472177705108, 3.56047666794643314415498003240, 5.51840139476854053978004396314, 6.75742646025671565045801544471, 7.74975081387417699648506717442, 8.491098854013019332631538598313, 9.194540561331585168417190505265, 9.863871208995538533908948882861, 11.34904394577793489782097136033, 11.64824143538440054982604406811

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