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Properties

Label 2-690-115.22-c1-0-20
Degree $2$
Conductor $690$
Sign $0.719 + 0.694i$
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  + (0.707 + 0.707i)2-s + (0.707 − 0.707i)3-s + 1.00i·4-s + (−0.643 − 2.14i)5-s + 1.00·6-s + (−2.01 + 2.01i)7-s + (−0.707 + 0.707i)8-s − 1.00i·9-s + (1.05 − 1.96i)10-s − 5.11i·11-s + (0.707 + 0.707i)12-s + (4.36 − 4.36i)13-s − 2.85·14-s + (−1.96 − 1.05i)15-s − 1.00·16-s + (0.390 − 0.390i)17-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + (0.408 − 0.408i)3-s + 0.500i·4-s + (−0.287 − 0.957i)5-s + 0.408·6-s + (−0.762 + 0.762i)7-s + (−0.250 + 0.250i)8-s − 0.333i·9-s + (0.335 − 0.622i)10-s − 1.54i·11-s + (0.204 + 0.204i)12-s + (1.21 − 1.21i)13-s − 0.762·14-s + (−0.508 − 0.273i)15-s − 0.250·16-s + (0.0948 − 0.0948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.719 + 0.694i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.719 + 0.694i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81155 - 0.732024i\)
\(L(\frac12)\) \(\approx\) \(1.81155 - 0.732024i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 + (-0.707 + 0.707i)T \)
5 \( 1 + (0.643 + 2.14i)T \)
23 \( 1 + (-0.204 + 4.79i)T \)
good7 \( 1 + (2.01 - 2.01i)T - 7iT^{2} \)
11 \( 1 + 5.11iT - 11T^{2} \)
13 \( 1 + (-4.36 + 4.36i)T - 13iT^{2} \)
17 \( 1 + (-0.390 + 0.390i)T - 17iT^{2} \)
19 \( 1 - 6.15T + 19T^{2} \)
29 \( 1 - 5.15iT - 29T^{2} \)
31 \( 1 - 5.55T + 31T^{2} \)
37 \( 1 + (1.66 - 1.66i)T - 37iT^{2} \)
41 \( 1 + 8.20T + 41T^{2} \)
43 \( 1 + (3.51 + 3.51i)T + 43iT^{2} \)
47 \( 1 + (3.69 + 3.69i)T + 47iT^{2} \)
53 \( 1 + (-3.27 - 3.27i)T + 53iT^{2} \)
59 \( 1 + 9.42iT - 59T^{2} \)
61 \( 1 - 6.90iT - 61T^{2} \)
67 \( 1 + (6.18 - 6.18i)T - 67iT^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + (9.69 - 9.69i)T - 73iT^{2} \)
79 \( 1 - 0.836T + 79T^{2} \)
83 \( 1 + (-7.68 - 7.68i)T + 83iT^{2} \)
89 \( 1 - 8.61T + 89T^{2} \)
97 \( 1 + (-7.13 + 7.13i)T - 97iT^{2} \)
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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

−10.31490857887866801641640642188, −9.043970700324554723667909803069, −8.512297601620518346997740584297, −7.992639414899097372491089595988, −6.66171505238833932488228391854, −5.80278716246295835263825630154, −5.18656171470716736396726937642, −3.54561463324071957204991073966, −3.03664096219944047303963959504, −0.899445881361417734586604445499, 1.79108736894056974017644587120, 3.21223148708904974873348859878, 3.82423885342942157799964763334, 4.73679157084213933476231882439, 6.24058949050726230402672064903, 6.98004687616988782171750473508, 7.78222618574282606137165858250, 9.269735065909244088135687900904, 9.899266728896433317876522603398, 10.42926507832822002154225672025

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