Properties

Label 2-405-1.1-c1-0-7
Degree $2$
Conductor $405$
Sign $1$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
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  + 2.08·2-s + 2.35·4-s − 5-s + 4.08·7-s + 0.734·8-s − 2.08·10-s + 1.35·11-s + 0.648·13-s + 8.52·14-s − 3.17·16-s + 1.35·17-s + 0.648·19-s − 2.35·20-s + 2.82·22-s − 4.79·23-s + 25-s + 1.35·26-s + 9.61·28-s − 3.87·29-s − 7.69·31-s − 8.08·32-s + 2.82·34-s − 4.08·35-s + 7.52·37-s + 1.35·38-s − 0.734·40-s + 0.179·41-s + ⋯
L(s)  = 1  + 1.47·2-s + 1.17·4-s − 0.447·5-s + 1.54·7-s + 0.259·8-s − 0.659·10-s + 0.407·11-s + 0.179·13-s + 2.27·14-s − 0.793·16-s + 0.327·17-s + 0.148·19-s − 0.525·20-s + 0.601·22-s − 0.998·23-s + 0.200·25-s + 0.265·26-s + 1.81·28-s − 0.719·29-s − 1.38·31-s − 1.42·32-s + 0.483·34-s − 0.690·35-s + 1.23·37-s + 0.219·38-s − 0.116·40-s + 0.0280·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(3.23394\)
Root analytic conductor: \(1.79831\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.985129585\)
\(L(\frac12)\) \(\approx\) \(2.985129585\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good2 \( 1 - 2.08T + 2T^{2} \)
7 \( 1 - 4.08T + 7T^{2} \)
11 \( 1 - 1.35T + 11T^{2} \)
13 \( 1 - 0.648T + 13T^{2} \)
17 \( 1 - 1.35T + 17T^{2} \)
19 \( 1 - 0.648T + 19T^{2} \)
23 \( 1 + 4.79T + 23T^{2} \)
29 \( 1 + 3.87T + 29T^{2} \)
31 \( 1 + 7.69T + 31T^{2} \)
37 \( 1 - 7.52T + 37T^{2} \)
41 \( 1 - 0.179T + 41T^{2} \)
43 \( 1 + 0.820T + 43T^{2} \)
47 \( 1 + 10.9T + 47T^{2} \)
53 \( 1 + 4.17T + 53T^{2} \)
59 \( 1 + 4.17T + 59T^{2} \)
61 \( 1 + 3.82T + 61T^{2} \)
67 \( 1 - 8.14T + 67T^{2} \)
71 \( 1 - 6.11T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 10.3T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + 13.5T + 97T^{2} \)
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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

−11.45359683933688728953493631087, −10.94163411239748998932598819198, −9.421856993621271807439932325262, −8.244814752793429897264647805926, −7.44243333622413619716615056519, −6.17699643520193400866159855186, −5.21494526850361755671440842486, −4.39699754656650699664459128108, −3.49752636174282218812187065946, −1.89342420448833102084428967755, 1.89342420448833102084428967755, 3.49752636174282218812187065946, 4.39699754656650699664459128108, 5.21494526850361755671440842486, 6.17699643520193400866159855186, 7.44243333622413619716615056519, 8.244814752793429897264647805926, 9.421856993621271807439932325262, 10.94163411239748998932598819198, 11.45359683933688728953493631087

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