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Properties

Label 2-690-69.68-c1-0-18
Degree $2$
Conductor $690$
Sign $-0.423 + 0.905i$
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 + (−1.71 − 0.251i)3-s − 4-s − 5-s + (−0.251 + 1.71i)6-s − 0.194i·7-s + i·8-s + (2.87 + 0.863i)9-s + i·10-s + 3.67·11-s + (1.71 + 0.251i)12-s + 3.85·13-s − 0.194·14-s + (1.71 + 0.251i)15-s + 16-s − 3.71·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.989 − 0.145i)3-s − 0.5·4-s − 0.447·5-s + (−0.102 + 0.699i)6-s − 0.0734i·7-s + 0.353i·8-s + (0.957 + 0.287i)9-s + 0.316i·10-s + 1.10·11-s + (0.494 + 0.0727i)12-s + 1.06·13-s − 0.0519·14-s + (0.442 + 0.0650i)15-s + 0.250·16-s − 0.900·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.423 + 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.423 + 0.905i$
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: \((2,\ 690,\ (\ :1/2),\ -0.423 + 0.905i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.473241 - 0.743627i\)
\(L(\frac12)\) \(\approx\) \(0.473241 - 0.743627i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.71 + 0.251i)T \)
5 \( 1 + T \)
23 \( 1 + (4.59 + 1.37i)T \)
good7 \( 1 + 0.194iT - 7T^{2} \)
11 \( 1 - 3.67T + 11T^{2} \)
13 \( 1 - 3.85T + 13T^{2} \)
17 \( 1 + 3.71T + 17T^{2} \)
19 \( 1 + 0.273iT - 19T^{2} \)
29 \( 1 + 9.11iT - 29T^{2} \)
31 \( 1 - 2.24T + 31T^{2} \)
37 \( 1 + 11.7iT - 37T^{2} \)
41 \( 1 + 9.65iT - 41T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 + 4.64iT - 47T^{2} \)
53 \( 1 - 4.09T + 53T^{2} \)
59 \( 1 - 2.49iT - 59T^{2} \)
61 \( 1 - 13.9iT - 61T^{2} \)
67 \( 1 + 10.9iT - 67T^{2} \)
71 \( 1 + 9.72iT - 71T^{2} \)
73 \( 1 - 11.0T + 73T^{2} \)
79 \( 1 + 6.43iT - 79T^{2} \)
83 \( 1 + 9.42T + 83T^{2} \)
89 \( 1 - 9.92T + 89T^{2} \)
97 \( 1 + 8.91iT - 97T^{2} \)
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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

−10.52080553374614991529271480626, −9.470356988529412775334327277356, −8.629445191654383848704507521781, −7.53890640126883408120661860972, −6.46172628525324702090017782074, −5.76198613877092615559250370041, −4.30820298416773476570959884172, −3.91346288941385523315445094525, −2.04972752470908295231638870224, −0.63110421680519600449659470390, 1.26196129404947055709915908416, 3.64125751921517369859918688313, 4.41816355558947379468523794559, 5.45533944765277825047165287176, 6.49706007502140185858037583651, 6.82814498603305316301164643481, 8.137691280684801301063036868366, 8.918398556308120284395635284745, 9.858111033191145493053983410425, 10.83764562909597558557733667565

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