Properties

Label 2-690-1.1-c1-0-8
Degree $2$
Conductor $690$
Sign $1$
Analytic cond. $5.50967$
Root an. cond. $2.34727$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 2·11-s + 12-s + 4·13-s + 15-s + 16-s + 6·17-s + 18-s − 8·19-s + 20-s − 2·22-s − 23-s + 24-s + 25-s + 4·26-s + 27-s + 4·29-s + 30-s + 32-s − 2·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.603·11-s + 0.288·12-s + 1.10·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s + 0.235·18-s − 1.83·19-s + 0.223·20-s − 0.426·22-s − 0.208·23-s + 0.204·24-s + 1/5·25-s + 0.784·26-s + 0.192·27-s + 0.742·29-s + 0.182·30-s + 0.176·32-s − 0.348·33-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(690\)    =    \(2 \cdot 3 \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(5.50967\)
Root analytic conductor: \(2.34727\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{690} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 690,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.015845354\)
\(L(\frac12)\) \(\approx\) \(3.015845354\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 - T \)
23 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 6 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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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

−10.46006842950388688821498725352, −9.778476790971364876210604546999, −8.514839073741154480133769062900, −8.012185614946042132726363856353, −6.73556129089751089343223996833, −5.99007031258734046981490521178, −4.96141389876974978877383998444, −3.84335009814648903472376632301, −2.88411126258645777238121403143, −1.64793340157193246414261963354, 1.64793340157193246414261963354, 2.88411126258645777238121403143, 3.84335009814648903472376632301, 4.96141389876974978877383998444, 5.99007031258734046981490521178, 6.73556129089751089343223996833, 8.012185614946042132726363856353, 8.514839073741154480133769062900, 9.778476790971364876210604546999, 10.46006842950388688821498725352

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