Properties

Label 2-690-15.2-c1-0-1
Degree $2$
Conductor $690$
Sign $-0.0387 - 0.999i$
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 + (−1.70 + 0.292i)3-s − 1.00i·4-s + (−1.73 − 1.41i)5-s + (−0.999 + 1.41i)6-s + (−2.44 − 2.44i)7-s + (−0.707 − 0.707i)8-s + (2.82 − i)9-s + (−2.22 + 0.224i)10-s + 5.65i·11-s + (0.292 + 1.70i)12-s + (1.44 − 1.44i)13-s − 3.46·14-s + (3.37 + 1.90i)15-s − 1.00·16-s + (−5.19 + 5.19i)17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.985 + 0.169i)3-s − 0.500i·4-s + (−0.774 − 0.632i)5-s + (−0.408 + 0.577i)6-s + (−0.925 − 0.925i)7-s + (−0.250 − 0.250i)8-s + (0.942 − 0.333i)9-s + (−0.703 + 0.0710i)10-s + 1.70i·11-s + (0.0845 + 0.492i)12-s + (0.402 − 0.402i)13-s − 0.925·14-s + (0.870 + 0.492i)15-s − 0.250·16-s + (−1.26 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0387 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $-0.0387 - 0.999i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ -0.0387 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.164965 + 0.171488i\)
\(L(\frac12)\) \(\approx\) \(0.164965 + 0.171488i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 0.292i)T \)
5 \( 1 + (1.73 + 1.41i)T \)
23 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + (2.44 + 2.44i)T + 7iT^{2} \)
11 \( 1 - 5.65iT - 11T^{2} \)
13 \( 1 + (-1.44 + 1.44i)T - 13iT^{2} \)
17 \( 1 + (5.19 - 5.19i)T - 17iT^{2} \)
19 \( 1 + 2.44iT - 19T^{2} \)
29 \( 1 - 9.12T + 29T^{2} \)
31 \( 1 + 4.89T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + (6.89 - 6.89i)T - 43iT^{2} \)
47 \( 1 + (2.04 - 2.04i)T - 47iT^{2} \)
53 \( 1 + (9.43 + 9.43i)T + 53iT^{2} \)
59 \( 1 + 8.34T + 59T^{2} \)
61 \( 1 + 8.89T + 61T^{2} \)
67 \( 1 + (1.10 + 1.10i)T + 67iT^{2} \)
71 \( 1 - 6.14iT - 71T^{2} \)
73 \( 1 + (-7.89 + 7.89i)T - 73iT^{2} \)
79 \( 1 + 0.449iT - 79T^{2} \)
83 \( 1 + (1.41 + 1.41i)T + 83iT^{2} \)
89 \( 1 + 2.19T + 89T^{2} \)
97 \( 1 + (-4.44 - 4.44i)T + 97iT^{2} \)
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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

−10.79352238624034286997593213052, −10.05532214556375652862400077635, −9.344585563938427033734187740525, −7.968594676415279823024945258617, −6.83522579380463680612979822597, −6.34363462430399986786011248356, −4.71557770029059530629093308613, −4.51108066876830105169121974798, −3.41805821829252120352112090655, −1.43396195901344587719508504781, 0.12463525596437182630034052012, 2.74252274766563989410714641979, 3.72331903374651836002115399120, 4.92203940575855037512542680658, 6.03603295701717321550803672636, 6.43381966764069566173491632042, 7.27715401005919476731994300646, 8.448452399095767249786548742312, 9.215537197891405595610873353309, 10.62620507137669044468378394332

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