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Properties

Label 2-17e2-17.15-c1-0-4
Degree $2$
Conductor $289$
Sign $-0.739 + 0.673i$
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  + (−1.70 − 1.70i)2-s + (−0.414 − i)3-s + 3.82i·4-s + (0.707 − 0.292i)5-s + (−1 + 2.41i)6-s + (2.41 + i)7-s + (3.12 − 3.12i)8-s + (1.29 − 1.29i)9-s + (−1.70 − 0.707i)10-s + (1 − 2.41i)11-s + (3.82 − 1.58i)12-s − 1.41i·13-s + (−2.41 − 5.82i)14-s + (−0.585 − 0.585i)15-s − 2.99·16-s + ⋯
L(s)  = 1  + (−1.20 − 1.20i)2-s + (−0.239 − 0.577i)3-s + 1.91i·4-s + (0.316 − 0.130i)5-s + (−0.408 + 0.985i)6-s + (0.912 + 0.377i)7-s + (1.10 − 1.10i)8-s + (0.430 − 0.430i)9-s + (−0.539 − 0.223i)10-s + (0.301 − 0.727i)11-s + (1.10 − 0.457i)12-s − 0.392i·13-s + (−0.645 − 1.55i)14-s + (−0.151 − 0.151i)15-s − 0.749·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.739 + 0.673i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.739 + 0.673i$
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: \((2,\ 289,\ (\ :1/2),\ -0.739 + 0.673i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.271490 - 0.701142i\)
\(L(\frac12)\) \(\approx\) \(0.271490 - 0.701142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 + 1.70i)T + 2iT^{2} \)
3 \( 1 + (0.414 + i)T + (-2.12 + 2.12i)T^{2} \)
5 \( 1 + (-0.707 + 0.292i)T + (3.53 - 3.53i)T^{2} \)
7 \( 1 + (-2.41 - i)T + (4.94 + 4.94i)T^{2} \)
11 \( 1 + (-1 + 2.41i)T + (-7.77 - 7.77i)T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
19 \( 1 + (-0.585 - 0.585i)T + 19iT^{2} \)
23 \( 1 + (-1.82 + 4.41i)T + (-16.2 - 16.2i)T^{2} \)
29 \( 1 + (-0.292 + 0.121i)T + (20.5 - 20.5i)T^{2} \)
31 \( 1 + (-3 - 7.24i)T + (-21.9 + 21.9i)T^{2} \)
37 \( 1 + (3.53 + 8.53i)T + (-26.1 + 26.1i)T^{2} \)
41 \( 1 + (1.12 + 0.464i)T + (28.9 + 28.9i)T^{2} \)
43 \( 1 + (0.585 - 0.585i)T - 43iT^{2} \)
47 \( 1 + 5.17iT - 47T^{2} \)
53 \( 1 + (1 + i)T + 53iT^{2} \)
59 \( 1 + (-4.24 + 4.24i)T - 59iT^{2} \)
61 \( 1 + (-3.53 - 1.46i)T + (43.1 + 43.1i)T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (-2.07 - 5i)T + (-50.2 + 50.2i)T^{2} \)
73 \( 1 + (-11.9 + 4.94i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (1.82 - 4.41i)T + (-55.8 - 55.8i)T^{2} \)
83 \( 1 + (-8.24 - 8.24i)T + 83iT^{2} \)
89 \( 1 - 6.58iT - 89T^{2} \)
97 \( 1 + (9.53 - 3.94i)T + (68.5 - 68.5i)T^{2} \)
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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

−11.35744839125333621501163461376, −10.60517901344709138590636844044, −9.599966456818817038509583909295, −8.707909674742857097325868997505, −8.007822428863121218514808848182, −6.80626963673156935839003315127, −5.38782320588269174138720625006, −3.62111211106019833427220480376, −2.11198360201484559726294194152, −0.980382722009355875930489929294, 1.63566644866198320671379263899, 4.35624260509463750072059396254, 5.31220489517812807633907312282, 6.52787799049123182009412936944, 7.47561759622645044163135159931, 8.170462251201709905868536595912, 9.435548459773123191366116116183, 9.930643406937352174356890668540, 10.81456568274008944841290721814, 11.77839884656162548735797852520

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