Properties

Label 2-3920-1.1-c1-0-27
Degree $2$
Conductor $3920$
Sign $1$
Analytic cond. $31.3013$
Root an. cond. $5.59476$
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  − 3·3-s − 5-s + 6·9-s + 5·11-s + 5·13-s + 3·15-s + 7·17-s − 2·19-s + 2·23-s + 25-s − 9·27-s + 7·29-s + 4·31-s − 15·33-s − 6·37-s − 15·39-s + 12·41-s + 2·43-s − 6·45-s + 47-s − 21·51-s − 5·55-s + 6·57-s − 4·59-s − 4·61-s − 5·65-s − 8·67-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s + 2·9-s + 1.50·11-s + 1.38·13-s + 0.774·15-s + 1.69·17-s − 0.458·19-s + 0.417·23-s + 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.718·31-s − 2.61·33-s − 0.986·37-s − 2.40·39-s + 1.87·41-s + 0.304·43-s − 0.894·45-s + 0.145·47-s − 2.94·51-s − 0.674·55-s + 0.794·57-s − 0.520·59-s − 0.512·61-s − 0.620·65-s − 0.977·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(31.3013\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.306903930\)
\(L(\frac12)\) \(\approx\) \(1.306903930\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
11 \( 1 - 5 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−8.469593510481410672158422929153, −7.53865014479941364859385861433, −6.76183869331890959102062964394, −6.14472686938251363416766925909, −5.73827754277642739938159369313, −4.68034948049666199716320051545, −4.09045006533871894101168233996, −3.23936592321491102560843324681, −1.34814974495374732009706645550, −0.867865270599308918807940906939, 0.867865270599308918807940906939, 1.34814974495374732009706645550, 3.23936592321491102560843324681, 4.09045006533871894101168233996, 4.68034948049666199716320051545, 5.73827754277642739938159369313, 6.14472686938251363416766925909, 6.76183869331890959102062964394, 7.53865014479941364859385861433, 8.469593510481410672158422929153

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