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Properties

Label 2-1935-1.1-c1-0-46
Degree $2$
Conductor $1935$
Sign $-1$
Analytic cond. $15.4510$
Root an. cond. $3.93078$
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  − 2-s − 4-s − 5-s + 4·7-s + 3·8-s + 10-s − 4·11-s + 2·13-s − 4·14-s − 16-s − 6·17-s − 6·19-s + 20-s + 4·22-s + 6·23-s + 25-s − 2·26-s − 4·28-s + 2·29-s − 4·31-s − 5·32-s + 6·34-s − 4·35-s + 8·37-s + 6·38-s − 3·40-s − 8·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 1.51·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s + 0.554·13-s − 1.06·14-s − 1/4·16-s − 1.45·17-s − 1.37·19-s + 0.223·20-s + 0.852·22-s + 1.25·23-s + 1/5·25-s − 0.392·26-s − 0.755·28-s + 0.371·29-s − 0.718·31-s − 0.883·32-s + 1.02·34-s − 0.676·35-s + 1.31·37-s + 0.973·38-s − 0.474·40-s − 1.24·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1935\)    =    \(3^{2} \cdot 5 \cdot 43\)
Sign: $-1$
Analytic conductor: \(15.4510\)
Root analytic conductor: \(3.93078\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1935,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
5 \( 1 + T \)
43 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−8.583249121456513011349194882063, −8.254141844044545683575126652738, −7.57388551453478509248538032114, −6.66232739555203435532058173801, −5.32753883250167872867290184035, −4.68368618297505069822092599172, −4.08671879573495977931383773399, −2.51981200347985481019084431547, −1.43256006482888552695616959253, 0, 1.43256006482888552695616959253, 2.51981200347985481019084431547, 4.08671879573495977931383773399, 4.68368618297505069822092599172, 5.32753883250167872867290184035, 6.66232739555203435532058173801, 7.57388551453478509248538032114, 8.254141844044545683575126652738, 8.583249121456513011349194882063

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