Properties

Label 2-690-15.8-c1-0-15
Degree $2$
Conductor $690$
Sign $0.749 + 0.662i$
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 − 2.23i·5-s + (0.999 + 1.41i)6-s + (0.707 − 0.707i)8-s + (2.82 + i)9-s + (−1.58 + 1.58i)10-s + 2.82i·11-s + (0.292 − 1.70i)12-s + (4.16 + 4.16i)13-s + (−0.654 + 3.81i)15-s − 1.00·16-s + (−3.65 − 3.65i)17-s + (−1.29 − 2.70i)18-s + 5.16i·19-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.985 − 0.169i)3-s + 0.500i·4-s − 0.999i·5-s + (0.408 + 0.577i)6-s + (0.250 − 0.250i)8-s + (0.942 + 0.333i)9-s + (−0.500 + 0.500i)10-s + 0.852i·11-s + (0.0845 − 0.492i)12-s + (1.15 + 1.15i)13-s + (−0.169 + 0.985i)15-s − 0.250·16-s + (−0.885 − 0.885i)17-s + (−0.304 − 0.638i)18-s + 1.18i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 690 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $0.749 + 0.662i$
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: \((2,\ 690,\ (\ :1/2),\ 0.749 + 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.793138 - 0.300122i\)
\(L(\frac12)\) \(\approx\) \(0.793138 - 0.300122i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
23 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 2.82iT - 11T^{2} \)
13 \( 1 + (-4.16 - 4.16i)T + 13iT^{2} \)
17 \( 1 + (3.65 + 3.65i)T + 17iT^{2} \)
19 \( 1 - 5.16iT - 19T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 - 10.3T + 31T^{2} \)
37 \( 1 + (-5.16 + 5.16i)T - 37iT^{2} \)
41 \( 1 + 6.11iT - 41T^{2} \)
43 \( 1 + (2.83 + 2.83i)T + 43iT^{2} \)
47 \( 1 + (-8.71 - 8.71i)T + 47iT^{2} \)
53 \( 1 + (-6.70 + 6.70i)T - 53iT^{2} \)
59 \( 1 + 9.89T + 59T^{2} \)
61 \( 1 - 8T + 61T^{2} \)
67 \( 1 + (-5.16 + 5.16i)T - 67iT^{2} \)
71 \( 1 - 3.05iT - 71T^{2} \)
73 \( 1 + (7.32 + 7.32i)T + 73iT^{2} \)
79 \( 1 - 3.16iT - 79T^{2} \)
83 \( 1 + (-5.88 + 5.88i)T - 83iT^{2} \)
89 \( 1 - 0.458T + 89T^{2} \)
97 \( 1 + (-1.16 + 1.16i)T - 97iT^{2} \)
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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

−10.40659444678048520085799293491, −9.552921900557789535247814905568, −8.832199136010943656062649961593, −7.84914418518137451769623933038, −6.82065013906330895846004458035, −5.95184332517292251467760929807, −4.65728336252795099705528580465, −4.13733923143049356280263859201, −2.07113355285060370684453634066, −0.958778701376155870771859703289, 0.872111323977647044875938444791, 2.88662618544675750812313032048, 4.21148002782277551353241201141, 5.48438889915690058936024480027, 6.34778465194740519791098965937, 6.69698863886307336456486957009, 7.982985938250441406383723191320, 8.706217301822226190150304070250, 9.995711589021767324216728304218, 10.58021227125029212408988893705

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