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Properties

Label 2-17e2-17.2-c1-0-3
Degree $2$
Conductor $289$
Sign $-0.832 - 0.553i$
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 + (−1 + 0.414i)3-s + 3.82i·4-s + (−0.292 − 0.707i)5-s + (−2.41 − i)6-s + (−1 + 2.41i)7-s + (−3.12 + 3.12i)8-s + (−1.29 + 1.29i)9-s + (0.707 − 1.70i)10-s + (2.41 + i)11-s + (−1.58 − 3.82i)12-s − 1.41i·13-s + (−5.82 + 2.41i)14-s + (0.585 + 0.585i)15-s − 2.99·16-s + ⋯
L(s)  = 1  + (1.20 + 1.20i)2-s + (−0.577 + 0.239i)3-s + 1.91i·4-s + (−0.130 − 0.316i)5-s + (−0.985 − 0.408i)6-s + (−0.377 + 0.912i)7-s + (−1.10 + 1.10i)8-s + (−0.430 + 0.430i)9-s + (0.223 − 0.539i)10-s + (0.727 + 0.301i)11-s + (−0.457 − 1.10i)12-s − 0.392i·13-s + (−1.55 + 0.645i)14-s + (0.151 + 0.151i)15-s − 0.749·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $-0.832 - 0.553i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ -0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.525093 + 1.73756i\)
\(L(\frac12)\) \(\approx\) \(0.525093 + 1.73756i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 0.414i)T + (2.12 - 2.12i)T^{2} \)
5 \( 1 + (0.292 + 0.707i)T + (-3.53 + 3.53i)T^{2} \)
7 \( 1 + (1 - 2.41i)T + (-4.94 - 4.94i)T^{2} \)
11 \( 1 + (-2.41 - i)T + (7.77 + 7.77i)T^{2} \)
13 \( 1 + 1.41iT - 13T^{2} \)
19 \( 1 + (0.585 + 0.585i)T + 19iT^{2} \)
23 \( 1 + (-4.41 - 1.82i)T + (16.2 + 16.2i)T^{2} \)
29 \( 1 + (0.121 + 0.292i)T + (-20.5 + 20.5i)T^{2} \)
31 \( 1 + (-7.24 + 3i)T + (21.9 - 21.9i)T^{2} \)
37 \( 1 + (8.53 - 3.53i)T + (26.1 - 26.1i)T^{2} \)
41 \( 1 + (-0.464 + 1.12i)T + (-28.9 - 28.9i)T^{2} \)
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 \( 1 + (1.46 - 3.53i)T + (-43.1 - 43.1i)T^{2} \)
67 \( 1 - 1.17T + 67T^{2} \)
71 \( 1 + (-5 + 2.07i)T + (50.2 - 50.2i)T^{2} \)
73 \( 1 + (4.94 + 11.9i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (4.41 + 1.82i)T + (55.8 + 55.8i)T^{2} \)
83 \( 1 + (8.24 + 8.24i)T + 83iT^{2} \)
89 \( 1 - 6.58iT - 89T^{2} \)
97 \( 1 + (-3.94 - 9.53i)T + (-68.5 + 68.5i)T^{2} \)
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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

−12.22249994635498271877099266316, −11.80756499389939265770816678795, −10.43560955136684727730861435460, −9.009359745549070946999504568497, −8.161762985081035208484906279379, −6.90504821037325140677386472411, −6.04541997972059184293274876409, −5.24748588045161929235453160516, −4.42807080742808074278823104632, −2.96098701911479966556842595915, 1.11634992736315779072119160441, 3.02379399210395561506730242619, 3.89994707217805160076139320815, 5.05760924249245720573450161156, 6.27782923678038741122707353322, 6.99980190141439052744357572349, 8.858261682857620519150896046242, 10.06179350234110939808724168027, 10.93929961447991861124095902787, 11.45566024978323192796655988157

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