Properties

Label 2-26520-1.1-c1-0-22
Degree $2$
Conductor $26520$
Sign $-1$
Analytic cond. $211.763$
Root an. cond. $14.5520$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 9-s + 4·11-s + 13-s − 15-s + 17-s − 4·19-s + 25-s − 27-s − 2·29-s + 8·31-s − 4·33-s − 10·37-s − 39-s + 10·41-s + 4·43-s + 45-s − 8·47-s − 7·49-s − 51-s − 2·53-s + 4·55-s + 4·57-s − 4·59-s + 6·61-s + 65-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.277·13-s − 0.258·15-s + 0.242·17-s − 0.917·19-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 1.43·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 1.56·41-s + 0.609·43-s + 0.149·45-s − 1.16·47-s − 49-s − 0.140·51-s − 0.274·53-s + 0.539·55-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.124·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(26520\)    =    \(2^{3} \cdot 3 \cdot 5 \cdot 13 \cdot 17\)
Sign: $-1$
Analytic conductor: \(211.763\)
Root analytic conductor: \(14.5520\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{26520} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 26520,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
5 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.74845288668895, −14.91032806670502, −14.48559294394924, −14.03313949771664, −13.37894600977612, −12.85098145421368, −12.28637695754305, −11.85109989357857, −11.17930590900665, −10.82445551201656, −10.06018174399319, −9.690718326677093, −9.000317408547430, −8.506955380894533, −7.844541977010261, −6.956710405092688, −6.627618774536191, −5.995274742661760, −5.564104344201259, −4.626895944567489, −4.247436412016212, −3.453781292130180, −2.645421576976926, −1.675353773179003, −1.170964794201668, 0, 1.170964794201668, 1.675353773179003, 2.645421576976926, 3.453781292130180, 4.247436412016212, 4.626895944567489, 5.564104344201259, 5.995274742661760, 6.627618774536191, 6.956710405092688, 7.844541977010261, 8.506955380894533, 9.000317408547430, 9.690718326677093, 10.06018174399319, 10.82445551201656, 11.17930590900665, 11.85109989357857, 12.28637695754305, 12.85098145421368, 13.37894600977612, 14.03313949771664, 14.48559294394924, 14.91032806670502, 15.74845288668895

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