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Properties

Label 2-17e2-17.15-c1-0-0
Degree $2$
Conductor $289$
Sign $0.611 - 0.790i$
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  + (−1.08 − 1.08i)2-s + (−0.515 − 1.24i)3-s + 0.347i·4-s + (−3.26 + 1.35i)5-s + (−0.789 + 1.90i)6-s + (0.320 + 0.132i)7-s + (−1.79 + 1.79i)8-s + (0.837 − 0.837i)9-s + (4.99 + 2.07i)10-s + (−0.673 + 1.62i)11-s + (0.432 − 0.179i)12-s + 3.29i·13-s + (−0.203 − 0.491i)14-s + (3.36 + 3.36i)15-s + 4.57·16-s + ⋯
L(s)  = 1  + (−0.766 − 0.766i)2-s + (−0.297 − 0.718i)3-s + 0.173i·4-s + (−1.45 + 0.604i)5-s + (−0.322 + 0.778i)6-s + (0.121 + 0.0502i)7-s + (−0.633 + 0.633i)8-s + (0.279 − 0.279i)9-s + (1.58 + 0.654i)10-s + (−0.202 + 0.489i)11-s + (0.124 − 0.0516i)12-s + 0.912i·13-s + (−0.0544 − 0.131i)14-s + (0.868 + 0.868i)15-s + 1.14·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.611 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.611 - 0.790i$
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: \((2,\ 289,\ (\ :1/2),\ 0.611 - 0.790i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.209739 + 0.102917i\)
\(L(\frac12)\) \(\approx\) \(0.209739 + 0.102917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.08 + 1.08i)T + 2iT^{2} \)
3 \( 1 + (0.515 + 1.24i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (3.26 - 1.35i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-0.320 - 0.132i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (0.673 - 1.62i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 - 3.29iT - 13T^{2} \)
19 \( 1 + (1.08 + 1.08i)T + 19iT^{2} \)
23 \( 1 + (1.07 - 2.60i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (1.09 - 0.453i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-2.71 - 6.56i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (-1.50 - 3.62i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (4.54 + 1.88i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (7.76 - 7.76i)T - 43iT^{2} \)
47 \( 1 - 5.12iT - 47T^{2} \)
53 \( 1 + (-5.91 - 5.91i)T + 53iT^{2} \)
59 \( 1 + (7.68 - 7.68i)T - 59iT^{2} \)
61 \( 1 + (4.07 + 1.68i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 + 8.07T + 67T^{2} \)
71 \( 1 + (4.30 + 10.3i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (1.57 - 0.651i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-4.43 + 10.7i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (1.51 + 1.51i)T + 83iT^{2} \)
89 \( 1 - 6.41iT - 89T^{2} \)
97 \( 1 + (-3.77 + 1.56i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.90200731219739935430710247196, −11.13769254129122085688853040211, −10.24393607658020324690556616569, −9.179097700814335226383428492504, −8.096057110288314256832897243644, −7.23482907386756631080627080937, −6.34878352347792897535552993343, −4.58004678671799076635894925083, −3.17711340227127519519961655926, −1.56999965799335895690120346417, 0.23990645694963760439887399147, 3.49351368575512815778562336862, 4.45185929166484228273688622257, 5.66621360995288010301046237982, 7.12698087563266745424371449104, 8.069578657422269594327326840352, 8.410186206547587785623089562345, 9.665809082114048291044205310318, 10.60168236568743653768516111060, 11.57216890568091732867960821689

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