Properties

Label 2-1290-1.1-c1-0-2
Degree $2$
Conductor $1290$
Sign $1$
Analytic cond. $10.3007$
Root an. cond. $3.20947$
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 + 6·11-s − 12-s + 2·13-s − 15-s + 16-s − 18-s − 2·19-s + 20-s − 6·22-s − 4·23-s + 24-s + 25-s − 2·26-s − 27-s + 2·29-s + 30-s + 8·31-s − 32-s − 6·33-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 + 1.80·11-s − 0.288·12-s + 0.554·13-s − 0.258·15-s + 1/4·16-s − 0.235·18-s − 0.458·19-s + 0.223·20-s − 1.27·22-s − 0.834·23-s + 0.204·24-s + 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.371·29-s + 0.182·30-s + 1.43·31-s − 0.176·32-s − 1.04·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1290\)    =    \(2 \cdot 3 \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(10.3007\)
Root analytic conductor: \(3.20947\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1290} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1290,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.204187850\)
\(L(\frac12)\) \(\approx\) \(1.204187850\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
43 \( 1 + T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−9.669316997866942035267106034843, −8.905441904315104023089774505851, −8.245815663689369566246874015568, −7.00513070761827742737411141354, −6.43835640630379150357644331002, −5.80425922759410663667601353410, −4.51557344602254348338728325856, −3.55092177583015585854210115686, −2.00952713752324090032312282495, −0.980459162915493741181211022221, 0.980459162915493741181211022221, 2.00952713752324090032312282495, 3.55092177583015585854210115686, 4.51557344602254348338728325856, 5.80425922759410663667601353410, 6.43835640630379150357644331002, 7.00513070761827742737411141354, 8.245815663689369566246874015568, 8.905441904315104023089774505851, 9.669316997866942035267106034843

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