Properties

Label 2-3900-1.1-c1-0-36
Degree $2$
Conductor $3900$
Sign $-1$
Analytic cond. $31.1416$
Root an. cond. $5.58047$
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 + 2·7-s + 9-s − 4·11-s − 13-s − 2·17-s − 2·19-s + 2·21-s + 27-s − 6·29-s − 10·31-s − 4·33-s − 10·37-s − 39-s + 8·41-s − 4·43-s + 4·47-s − 3·49-s − 2·51-s + 10·53-s − 2·57-s − 8·59-s − 14·61-s + 2·63-s − 2·67-s + 16·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.277·13-s − 0.485·17-s − 0.458·19-s + 0.436·21-s + 0.192·27-s − 1.11·29-s − 1.79·31-s − 0.696·33-s − 1.64·37-s − 0.160·39-s + 1.24·41-s − 0.609·43-s + 0.583·47-s − 3/7·49-s − 0.280·51-s + 1.37·53-s − 0.264·57-s − 1.04·59-s − 1.79·61-s + 0.251·63-s − 0.244·67-s + 1.89·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3900\)    =    \(2^{2} \cdot 3 \cdot 5^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(31.1416\)
Root analytic conductor: \(5.58047\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{3900} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 3900,\ (\ :1/2),\ -1)\)

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 \( 1 \)
13 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
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

−8.007242089652707299494642405749, −7.55475189997539123676200943441, −6.85607792524228677279435904172, −5.68556607312720044752967696384, −5.11884292421876510830098167220, −4.26937900663124280279842805225, −3.39610928208162456611102744465, −2.36732616538951941347129585944, −1.71945211223746090104912796041, 0, 1.71945211223746090104912796041, 2.36732616538951941347129585944, 3.39610928208162456611102744465, 4.26937900663124280279842805225, 5.11884292421876510830098167220, 5.68556607312720044752967696384, 6.85607792524228677279435904172, 7.55475189997539123676200943441, 8.007242089652707299494642405749

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