Properties

Label 2-690-69.68-c1-0-17
Degree $2$
Conductor $690$
Sign $0.493 - 0.869i$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
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  + i·2-s + (−0.597 + 1.62i)3-s − 4-s + 5-s + (−1.62 − 0.597i)6-s − 3.80i·7-s i·8-s + (−2.28 − 1.94i)9-s + i·10-s + 3.48·11-s + (0.597 − 1.62i)12-s + 3.19·13-s + 3.80·14-s + (−0.597 + 1.62i)15-s + 16-s + 4.76·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.344 + 0.938i)3-s − 0.5·4-s + 0.447·5-s + (−0.663 − 0.243i)6-s − 1.43i·7-s − 0.353i·8-s + (−0.762 − 0.647i)9-s + 0.316i·10-s + 1.05·11-s + (0.172 − 0.469i)12-s + 0.887·13-s + 1.01·14-s + (−0.154 + 0.419i)15-s + 0.250·16-s + 1.15·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.493 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.493 - 0.869i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.493 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27673 + 0.743234i\)
\(L(\frac12)\) \(\approx\) \(1.27673 + 0.743234i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.597 - 1.62i)T \)
5 \( 1 - T \)
23 \( 1 + (0.785 - 4.73i)T \)
good7 \( 1 + 3.80iT - 7T^{2} \)
11 \( 1 - 3.48T + 11T^{2} \)
13 \( 1 - 3.19T + 13T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 - 1.88iT - 19T^{2} \)
29 \( 1 + 7.50iT - 29T^{2} \)
31 \( 1 + 3.72T + 31T^{2} \)
37 \( 1 - 4.88iT - 37T^{2} \)
41 \( 1 - 3.99iT - 41T^{2} \)
43 \( 1 + 6.96iT - 43T^{2} \)
47 \( 1 + 10.8iT - 47T^{2} \)
53 \( 1 - 7.07T + 53T^{2} \)
59 \( 1 - 7.35iT - 59T^{2} \)
61 \( 1 - 3.31iT - 61T^{2} \)
67 \( 1 + 3.70iT - 67T^{2} \)
71 \( 1 - 9.92iT - 71T^{2} \)
73 \( 1 - 3.18T + 73T^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 - 11.0T + 89T^{2} \)
97 \( 1 - 2.98iT - 97T^{2} \)
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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

−10.27639579467512536826251539734, −9.892939138186215220695343672612, −8.984993916156491066891093034278, −7.989784867378062328006446554028, −6.96136340056219602581744932160, −6.09235729548157655915273313898, −5.31158641419261477547347900056, −3.99287019926624776364639031332, −3.66691091851087366796110374673, −1.07696770045772550266224497628, 1.24011701488949410314518026161, 2.27580669699279282432018471717, 3.38923427573197413764750730317, 5.03258156674706163770201247041, 5.88397082582518152258211743064, 6.53631691020063481465734647499, 7.84981553383223947348695449105, 8.888284166615562724552572464292, 9.206204986578723530113504745037, 10.57958196928520766951277835460

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