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Properties

Label 2-690-345.344-c1-0-4
Degree $2$
Conductor $690$
Sign $-0.876 - 0.481i$
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  − 2-s + (−1.29 + 1.15i)3-s + 4-s + (2.22 − 0.223i)5-s + (1.29 − 1.15i)6-s − 0.666·7-s − 8-s + (0.348 − 2.97i)9-s + (−2.22 + 0.223i)10-s − 4.92·11-s + (−1.29 + 1.15i)12-s + 3.71i·13-s + 0.666·14-s + (−2.62 + 2.85i)15-s + 16-s + 0.810i·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.747 + 0.664i)3-s + 0.5·4-s + (0.994 − 0.100i)5-s + (0.528 − 0.470i)6-s − 0.252·7-s − 0.353·8-s + (0.116 − 0.993i)9-s + (−0.703 + 0.0707i)10-s − 1.48·11-s + (−0.373 + 0.332i)12-s + 1.02i·13-s + 0.178·14-s + (−0.676 + 0.736i)15-s + 0.250·16-s + 0.196i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.876 - 0.481i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (689, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.876 - 0.481i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.113975 + 0.444498i\)
\(L(\frac12)\) \(\approx\) \(0.113975 + 0.444498i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.29 - 1.15i)T \)
5 \( 1 + (-2.22 + 0.223i)T \)
23 \( 1 + (1.14 + 4.65i)T \)
good7 \( 1 + 0.666T + 7T^{2} \)
11 \( 1 + 4.92T + 11T^{2} \)
13 \( 1 - 3.71iT - 13T^{2} \)
17 \( 1 - 0.810iT - 17T^{2} \)
19 \( 1 - 4.16iT - 19T^{2} \)
29 \( 1 - 9.02iT - 29T^{2} \)
31 \( 1 + 6.13T + 31T^{2} \)
37 \( 1 + 2.21T + 37T^{2} \)
41 \( 1 - 2.54iT - 41T^{2} \)
43 \( 1 - 3.56T + 43T^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 - 10.6iT - 53T^{2} \)
59 \( 1 - 6.81iT - 59T^{2} \)
61 \( 1 + 9.21iT - 61T^{2} \)
67 \( 1 + 7.95T + 67T^{2} \)
71 \( 1 - 14.5iT - 71T^{2} \)
73 \( 1 - 4.84iT - 73T^{2} \)
79 \( 1 + 11.5iT - 79T^{2} \)
83 \( 1 + 13.5iT - 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 1.04T + 97T^{2} \)
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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

−10.51772083021033126305245873491, −10.12667449601479672887567260646, −9.268999537380519475305419640087, −8.536711708072737853602659648600, −7.23950757388970952952408230825, −6.30807476547928706702965285250, −5.56901629949894359610665583891, −4.63044839835692625375435136629, −3.10801634011309681170060995705, −1.69248628400593454108154192910, 0.31493194169085131987149106943, 1.95214538877243704051158864616, 2.92311888366755439722639021178, 5.12167469639706788541602545126, 5.66566157124678439989280117362, 6.59784558107928633798772280422, 7.53917223053404526078240336344, 8.177358025084413158744579227030, 9.465933572688737082939838550383, 10.15352738991417328748052510420

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