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Properties

Label 2-690-345.344-c1-0-32
Degree $2$
Conductor $690$
Sign $0.704 + 0.709i$
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 + (−1.68 − 0.401i)3-s + 4-s + (1.83 − 1.28i)5-s + (−1.68 − 0.401i)6-s + 1.73·7-s + 8-s + (2.67 + 1.35i)9-s + (1.83 − 1.28i)10-s − 1.14·11-s + (−1.68 − 0.401i)12-s − 5.69i·13-s + 1.73·14-s + (−3.59 + 1.42i)15-s + 16-s + 5.08i·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.972 − 0.231i)3-s + 0.5·4-s + (0.818 − 0.574i)5-s + (−0.687 − 0.163i)6-s + 0.656·7-s + 0.353·8-s + (0.892 + 0.450i)9-s + (0.578 − 0.406i)10-s − 0.344·11-s + (−0.486 − 0.115i)12-s − 1.57i·13-s + 0.463·14-s + (−0.929 + 0.368i)15-s + 0.250·16-s + 1.23i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.704 + 0.709i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.704 + 0.709i$
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: \((2,\ 690,\ (\ :1/2),\ 0.704 + 0.709i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.93187 - 0.803917i\)
\(L(\frac12)\) \(\approx\) \(1.93187 - 0.803917i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.68 + 0.401i)T \)
5 \( 1 + (-1.83 + 1.28i)T \)
23 \( 1 + (1.88 + 4.40i)T \)
good7 \( 1 - 1.73T + 7T^{2} \)
11 \( 1 + 1.14T + 11T^{2} \)
13 \( 1 + 5.69iT - 13T^{2} \)
17 \( 1 - 5.08iT - 17T^{2} \)
19 \( 1 - 4.40iT - 19T^{2} \)
29 \( 1 + 4.89iT - 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
37 \( 1 - 11.5T + 37T^{2} \)
41 \( 1 + 5.25iT - 41T^{2} \)
43 \( 1 + 3.19T + 43T^{2} \)
47 \( 1 + 2.36T + 47T^{2} \)
53 \( 1 - 7.04iT - 53T^{2} \)
59 \( 1 + 1.33iT - 59T^{2} \)
61 \( 1 + 7.49iT - 61T^{2} \)
67 \( 1 + 15.5T + 67T^{2} \)
71 \( 1 - 13.2iT - 71T^{2} \)
73 \( 1 - 4.45iT - 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 - 4.09iT - 83T^{2} \)
89 \( 1 - 2.58T + 89T^{2} \)
97 \( 1 + 4.22T + 97T^{2} \)
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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

−10.31815423894448396280856664725, −10.05548701905349755628164366434, −8.259260365238594230074606170875, −7.84068745465800874098147280946, −6.25070354715727504396443105206, −5.90042879814720867621432222830, −5.02447403966154100238629621483, −4.18387285179009158362758297420, −2.42380215480678717034747336744, −1.14410478378647753456670856014, 1.61031503808619210587586946756, 2.92659075110781573265460652036, 4.51931274987304744661182350997, 4.98548644946427454521030102157, 6.09326833147497238279286711551, 6.75321163791235204411148776121, 7.55356074302763037616957989821, 9.192309725436583744296446870454, 9.846850472224021226750517936146, 10.83234272227837424860013407055

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