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Properties

Label 2-290-145.144-c1-0-11
Degree $2$
Conductor $290$
Sign $0.132 + 0.991i$
Analytic cond. $2.31566$
Root an. cond. $1.52172$
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-s − 0.367·3-s + 4-s + (1.82 − 1.29i)5-s + 0.367·6-s − 2.99i·7-s − 8-s − 2.86·9-s + (−1.82 + 1.29i)10-s − 1.08i·11-s − 0.367·12-s + 1.37i·13-s + 2.99i·14-s + (−0.671 + 0.474i)15-s + 16-s + 0.212·17-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.212·3-s + 0.5·4-s + (0.816 − 0.576i)5-s + 0.150·6-s − 1.13i·7-s − 0.353·8-s − 0.954·9-s + (−0.577 + 0.407i)10-s − 0.328i·11-s − 0.106·12-s + 0.381i·13-s + 0.800i·14-s + (−0.173 + 0.122i)15-s + 0.250·16-s + 0.0514·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.132 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(290\)    =    \(2 \cdot 5 \cdot 29\)
Sign: $0.132 + 0.991i$
Analytic conductor: \(2.31566\)
Root analytic conductor: \(1.52172\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{290} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 290,\ (\ :1/2),\ 0.132 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.647549 - 0.566822i\)
\(L(\frac12)\) \(\approx\) \(0.647549 - 0.566822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 + (-1.82 + 1.29i)T \)
29 \( 1 + (-2.49 + 4.77i)T \)
good3 \( 1 + 0.367T + 3T^{2} \)
7 \( 1 + 2.99iT - 7T^{2} \)
11 \( 1 + 1.08iT - 11T^{2} \)
13 \( 1 - 1.37iT - 13T^{2} \)
17 \( 1 - 0.212T + 17T^{2} \)
19 \( 1 + 6.70iT - 19T^{2} \)
23 \( 1 - 1.06iT - 23T^{2} \)
31 \( 1 + 4.22iT - 31T^{2} \)
37 \( 1 + 5.72T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 - 6.38T + 47T^{2} \)
53 \( 1 - 10.0iT - 53T^{2} \)
59 \( 1 + 5.28T + 59T^{2} \)
61 \( 1 - 3.38iT - 61T^{2} \)
67 \( 1 + 10.2iT - 67T^{2} \)
71 \( 1 + 5.04T + 71T^{2} \)
73 \( 1 - 3.51T + 73T^{2} \)
79 \( 1 - 8.85iT - 79T^{2} \)
83 \( 1 - 12.4iT - 83T^{2} \)
89 \( 1 - 2.95iT - 89T^{2} \)
97 \( 1 + 16.0T + 97T^{2} \)
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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

−11.28963513008889981483110204438, −10.67117935974092545478227118523, −9.578224413202366087279501393899, −8.900436459372365368412246868576, −7.83500915007048640119504076535, −6.68724773420441089158934972771, −5.73990482455273332912497868679, −4.43911639623123619597724518165, −2.63907201912815907265136640822, −0.833987592404233600089786474684, 2.01666477860100381709347468096, 3.14734344746912834847124415031, 5.45377011019652203525098385428, 5.96821580463601350395530094190, 7.14881999930121473340304843052, 8.455951638064548399870273991987, 9.098706740663143011489319631608, 10.21375252254506214225932155853, 10.81907446763011072663952975608, 12.01396576552712369837294241926

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