Properties

Label 2-9405-1.1-c1-0-134
Degree $2$
Conductor $9405$
Sign $1$
Analytic cond. $75.0993$
Root an. cond. $8.66598$
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  + 2.39·2-s + 3.74·4-s − 5-s − 2.89·7-s + 4.19·8-s − 2.39·10-s + 11-s + 4.73·13-s − 6.93·14-s + 2.56·16-s + 5.65·17-s + 19-s − 3.74·20-s + 2.39·22-s + 4.00·23-s + 25-s + 11.3·26-s − 10.8·28-s − 9.32·29-s − 6.60·31-s − 2.24·32-s + 13.5·34-s + 2.89·35-s + 6.07·37-s + 2.39·38-s − 4.19·40-s − 5.47·41-s + ⋯
L(s)  = 1  + 1.69·2-s + 1.87·4-s − 0.447·5-s − 1.09·7-s + 1.48·8-s − 0.758·10-s + 0.301·11-s + 1.31·13-s − 1.85·14-s + 0.640·16-s + 1.37·17-s + 0.229·19-s − 0.838·20-s + 0.511·22-s + 0.835·23-s + 0.200·25-s + 2.22·26-s − 2.05·28-s − 1.73·29-s − 1.18·31-s − 0.397·32-s + 2.32·34-s + 0.489·35-s + 0.999·37-s + 0.388·38-s − 0.663·40-s − 0.854·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9405\)    =    \(3^{2} \cdot 5 \cdot 11 \cdot 19\)
Sign: $1$
Analytic conductor: \(75.0993\)
Root analytic conductor: \(8.66598\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.447718874\)
\(L(\frac12)\) \(\approx\) \(5.447718874\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
11 \( 1 - T \)
19 \( 1 - T \)
good2 \( 1 - 2.39T + 2T^{2} \)
7 \( 1 + 2.89T + 7T^{2} \)
13 \( 1 - 4.73T + 13T^{2} \)
17 \( 1 - 5.65T + 17T^{2} \)
23 \( 1 - 4.00T + 23T^{2} \)
29 \( 1 + 9.32T + 29T^{2} \)
31 \( 1 + 6.60T + 31T^{2} \)
37 \( 1 - 6.07T + 37T^{2} \)
41 \( 1 + 5.47T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 0.295T + 47T^{2} \)
53 \( 1 - 3.81T + 53T^{2} \)
59 \( 1 - 5.54T + 59T^{2} \)
61 \( 1 + 1.01T + 61T^{2} \)
67 \( 1 - 6.98T + 67T^{2} \)
71 \( 1 - 1.02T + 71T^{2} \)
73 \( 1 + 0.202T + 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 15.8T + 89T^{2} \)
97 \( 1 - 4.81T + 97T^{2} \)
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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

−7.33908764996712674220205317201, −6.87204956272729777408502061590, −5.99287652759232128939669207394, −5.74988290362478611505772049188, −4.96128856330064712339818867161, −3.92940805541184765348540977539, −3.57190785743125325960999159437, −3.16317972728223342266884101245, −2.07283915293441647267145009974, −0.869743060948438292399171644980, 0.869743060948438292399171644980, 2.07283915293441647267145009974, 3.16317972728223342266884101245, 3.57190785743125325960999159437, 3.92940805541184765348540977539, 4.96128856330064712339818867161, 5.74988290362478611505772049188, 5.99287652759232128939669207394, 6.87204956272729777408502061590, 7.33908764996712674220205317201

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