Properties

Label 2-690-15.8-c1-0-8
Degree $2$
Conductor $690$
Sign $0.955 - 0.296i$
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 + (−0.292 − 1.70i)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 + (1.70 − 0.292i)12-s + (1.44 + 1.44i)13-s + 3.46·14-s + (−2.92 − 2.54i)15-s − 1.00·16-s + (5.19 + 5.19i)17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.169 − 0.985i)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 + (0.492 − 0.0845i)12-s + (0.402 + 0.402i)13-s + 0.925·14-s + (−0.754 − 0.656i)15-s − 0.250·16-s + (1.26 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.955 - 0.296i$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{690} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 0.955 - 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883833 + 0.134059i\)
\(L(\frac12)\) \(\approx\) \(0.883833 + 0.134059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.292 + 1.70i)T \)
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

−10.24945924649842824597613661331, −9.651568728135258520196617924334, −8.913518066654358977119478291484, −8.018534731483481956581701533955, −7.04709901523147797739832943695, −6.05187916246029503543237123211, −5.37642035257147111108559909976, −3.71465378532330843198774169791, −2.20774796845626966531739897226, −1.58628915072729756364566966460, 0.57248737610152807230572315066, 3.01827393744987360127559531679, 3.66168473135700658217784601471, 5.37978903365243093009342005430, 5.87355182175439583576410389270, 6.81840367824518165057336444602, 7.78973283708739533078522303669, 9.073653304572898345469384779968, 9.494602976278124536477753713132, 10.43432308149138340699328829973

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