Properties

Label 2-1035-1.1-c1-0-9
Degree $2$
Conductor $1035$
Sign $1$
Analytic cond. $8.26451$
Root an. cond. $2.87480$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.44·2-s + 3.99·4-s + 5-s − 7-s − 4.89·8-s − 2.44·10-s + 2.44·11-s − 0.449·13-s + 2.44·14-s + 3.99·16-s + 0.550·17-s − 0.449·19-s + 3.99·20-s − 5.99·22-s + 23-s + 25-s + 1.10·26-s − 3.99·28-s + 4.34·29-s + 9.89·31-s − 1.34·34-s − 35-s − 5.89·37-s + 1.10·38-s − 4.89·40-s − 0.550·41-s + 2·43-s + ⋯
L(s)  = 1  − 1.73·2-s + 1.99·4-s + 0.447·5-s − 0.377·7-s − 1.73·8-s − 0.774·10-s + 0.738·11-s − 0.124·13-s + 0.654·14-s + 0.999·16-s + 0.133·17-s − 0.103·19-s + 0.894·20-s − 1.27·22-s + 0.208·23-s + 0.200·25-s + 0.215·26-s − 0.755·28-s + 0.807·29-s + 1.77·31-s − 0.231·34-s − 0.169·35-s − 0.969·37-s + 0.178·38-s − 0.774·40-s − 0.0859·41-s + 0.304·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1035 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1035\)    =    \(3^{2} \cdot 5 \cdot 23\)
Sign: $1$
Analytic conductor: \(8.26451\)
Root analytic conductor: \(2.87480\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1035,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7510636158\)
\(L(\frac12)\) \(\approx\) \(0.7510636158\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 - T \)
good2 \( 1 + 2.44T + 2T^{2} \)
7 \( 1 + T + 7T^{2} \)
11 \( 1 - 2.44T + 11T^{2} \)
13 \( 1 + 0.449T + 13T^{2} \)
17 \( 1 - 0.550T + 17T^{2} \)
19 \( 1 + 0.449T + 19T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 - 9.89T + 31T^{2} \)
37 \( 1 + 5.89T + 37T^{2} \)
41 \( 1 + 0.550T + 41T^{2} \)
43 \( 1 - 2T + 43T^{2} \)
47 \( 1 + 3.55T + 47T^{2} \)
53 \( 1 + 5.44T + 53T^{2} \)
59 \( 1 - 4.34T + 59T^{2} \)
61 \( 1 - 15.3T + 61T^{2} \)
67 \( 1 + 7T + 67T^{2} \)
71 \( 1 + 10.3T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 - 9.24T + 83T^{2} \)
89 \( 1 - 7.10T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−10.04208356181078830870213166871, −9.054712595090912563085757480082, −8.538659652094765271947710581392, −7.62233718444430191573123124251, −6.67731781997491447287010280237, −6.21774856444065041237593704809, −4.77009662969871197925817999419, −3.22685565937445748552169871403, −2.07150504732527634756853433338, −0.883812992174290884525133007365, 0.883812992174290884525133007365, 2.07150504732527634756853433338, 3.22685565937445748552169871403, 4.77009662969871197925817999419, 6.21774856444065041237593704809, 6.67731781997491447287010280237, 7.62233718444430191573123124251, 8.538659652094765271947710581392, 9.054712595090912563085757480082, 10.04208356181078830870213166871

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