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Properties

Label 2-1935-215.214-c0-0-4
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  + 1.80·2-s + 2.24·4-s − 5-s − 0.445·7-s + 2.24·8-s − 1.80·10-s + 1.80·11-s − 0.801·14-s + 1.80·16-s − 2.24·20-s + 3.24·22-s + 25-s − 28-s − 0.445·31-s + 1.00·32-s + 0.445·35-s − 1.80·37-s − 2.24·40-s − 1.24·41-s + 43-s + 4.04·44-s − 0.801·49-s + 1.80·50-s − 1.80·55-s − 1.00·56-s − 1.24·59-s − 0.801·62-s + ⋯
L(s)  = 1  + 1.80·2-s + 2.24·4-s − 5-s − 0.445·7-s + 2.24·8-s − 1.80·10-s + 1.80·11-s − 0.801·14-s + 1.80·16-s − 2.24·20-s + 3.24·22-s + 25-s − 28-s − 0.445·31-s + 1.00·32-s + 0.445·35-s − 1.80·37-s − 2.24·40-s − 1.24·41-s + 43-s + 4.04·44-s − 0.801·49-s + 1.80·50-s − 1.80·55-s − 1.00·56-s − 1.24·59-s − 0.801·62-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

\(L(\frac{1}{2})\) \(\approx\) \(2.909636061\)
\(L(\frac12)\) \(\approx\) \(2.909636061\)
\(L(1)\) 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 \( 1 - 1.80T + T^{2} \)
7 \( 1 + 0.445T + T^{2} \)
11 \( 1 - 1.80T + T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 0.445T + T^{2} \)
37 \( 1 + 1.80T + T^{2} \)
41 \( 1 + 1.24T + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 1.24T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 1.24T + T^{2} \)
79 \( 1 + 1.80T + T^{2} \)
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

−9.342460449536754255150905776498, −8.514386529771318689179403797686, −7.35299113478348421993566099425, −6.79300620645872895825538857516, −6.19273353193436452048808011160, −5.16768815058766653646979530719, −4.31635331839078558348132888868, −3.68627566351996704290614547565, −3.11861089773221969908562269658, −1.66038129199918641910792858150, 1.66038129199918641910792858150, 3.11861089773221969908562269658, 3.68627566351996704290614547565, 4.31635331839078558348132888868, 5.16768815058766653646979530719, 6.19273353193436452048808011160, 6.79300620645872895825538857516, 7.35299113478348421993566099425, 8.514386529771318689179403797686, 9.342460449536754255150905776498

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