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Properties

Label 2-1935-215.214-c0-0-0
Degree $2$
Conductor $1935$
Sign $1$
Analytic cond. $0.965690$
Root an. cond. $0.982695$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.24·2-s + 0.554·4-s − 5-s − 1.80·7-s + 0.554·8-s + 1.24·10-s − 1.24·11-s + 2.24·14-s − 1.24·16-s − 0.554·20-s + 1.55·22-s + 25-s − 0.999·28-s − 1.80·31-s + 0.999·32-s + 1.80·35-s + 1.24·37-s − 0.554·40-s + 0.445·41-s + 43-s − 0.692·44-s + 2.24·49-s − 1.24·50-s + 1.24·55-s − 1.00·56-s + 0.445·59-s + 2.24·62-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.554·4-s − 5-s − 1.80·7-s + 0.554·8-s + 1.24·10-s − 1.24·11-s + 2.24·14-s − 1.24·16-s − 0.554·20-s + 1.55·22-s + 25-s − 0.999·28-s − 1.80·31-s + 0.999·32-s + 1.80·35-s + 1.24·37-s − 0.554·40-s + 0.445·41-s + 43-s − 0.692·44-s + 2.24·49-s − 1.24·50-s + 1.24·55-s − 1.00·56-s + 0.445·59-s + 2.24·62-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1935\)    =    \(3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(0.965690\)
Root analytic conductor: \(0.982695\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1935} (1504, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1935,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2349964447\)
\(L(\frac12)\) \(\approx\) \(0.2349964447\)
\(L(1)\) 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 \( 1 + 1.24T + T^{2} \)
7 \( 1 + 1.80T + T^{2} \)
11 \( 1 + 1.24T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.80T + T^{2} \)
37 \( 1 - 1.24T + T^{2} \)
41 \( 1 - 0.445T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - 0.445T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + 0.445T + T^{2} \)
79 \( 1 - 1.24T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.319792738420322926559584120818, −8.750689893090722414717218764320, −7.75114933546334833167901028971, −7.40795113109321796318584853575, −6.54270866810479516543542239325, −5.50696207723491975944100191041, −4.31492381973894950064962589631, −3.39720372048002455713321503097, −2.44397028922928930563080666585, −0.54741210409154972185318167028, 0.54741210409154972185318167028, 2.44397028922928930563080666585, 3.39720372048002455713321503097, 4.31492381973894950064962589631, 5.50696207723491975944100191041, 6.54270866810479516543542239325, 7.40795113109321796318584853575, 7.75114933546334833167901028971, 8.750689893090722414717218764320, 9.319792738420322926559584120818

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