Properties

Label 2-405-1.1-c3-0-5
Degree $2$
Conductor $405$
Sign $1$
Analytic cond. $23.8957$
Root an. cond. $4.88833$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732·2-s − 7.46·4-s − 5·5-s − 6.92·7-s − 11.3·8-s − 3.66·10-s − 37.4·11-s − 38.9·13-s − 5.07·14-s + 51.4·16-s + 80.9·17-s + 112.·19-s + 37.3·20-s − 27.4·22-s − 13.4·23-s + 25·25-s − 28.4·26-s + 51.7·28-s + 43.4·29-s − 149.·31-s + 128.·32-s + 59.2·34-s + 34.6·35-s − 218.·37-s + 82.5·38-s + 56.6·40-s + 372.·41-s + ⋯
L(s)  = 1  + 0.258·2-s − 0.933·4-s − 0.447·5-s − 0.374·7-s − 0.500·8-s − 0.115·10-s − 1.02·11-s − 0.830·13-s − 0.0968·14-s + 0.803·16-s + 1.15·17-s + 1.36·19-s + 0.417·20-s − 0.266·22-s − 0.121·23-s + 0.200·25-s − 0.214·26-s + 0.349·28-s + 0.278·29-s − 0.867·31-s + 0.708·32-s + 0.299·34-s + 0.167·35-s − 0.970·37-s + 0.352·38-s + 0.223·40-s + 1.41·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $1$
Analytic conductor: \(23.8957\)
Root analytic conductor: \(4.88833\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 405,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.120472273\)
\(L(\frac12)\) \(\approx\) \(1.120472273\)
\(L(\frac{5}{2})\) 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 \( 1 + 5T \)
good2 \( 1 - 0.732T + 8T^{2} \)
7 \( 1 + 6.92T + 343T^{2} \)
11 \( 1 + 37.4T + 1.33e3T^{2} \)
13 \( 1 + 38.9T + 2.19e3T^{2} \)
17 \( 1 - 80.9T + 4.91e3T^{2} \)
19 \( 1 - 112.T + 6.85e3T^{2} \)
23 \( 1 + 13.4T + 1.21e4T^{2} \)
29 \( 1 - 43.4T + 2.43e4T^{2} \)
31 \( 1 + 149.T + 2.97e4T^{2} \)
37 \( 1 + 218.T + 5.06e4T^{2} \)
41 \( 1 - 372.T + 6.89e4T^{2} \)
43 \( 1 - 460.T + 7.95e4T^{2} \)
47 \( 1 - 214.T + 1.03e5T^{2} \)
53 \( 1 - 445.T + 1.48e5T^{2} \)
59 \( 1 - 401.T + 2.05e5T^{2} \)
61 \( 1 - 1.44T + 2.26e5T^{2} \)
67 \( 1 + 816.T + 3.00e5T^{2} \)
71 \( 1 - 147.T + 3.57e5T^{2} \)
73 \( 1 - 432.T + 3.89e5T^{2} \)
79 \( 1 - 384.T + 4.93e5T^{2} \)
83 \( 1 - 1.26e3T + 5.71e5T^{2} \)
89 \( 1 - 513T + 7.04e5T^{2} \)
97 \( 1 + 1.09e3T + 9.12e5T^{2} \)
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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

−10.71219876121537935078554994302, −9.841003726026216146639662146285, −9.139696903754866133881869430916, −7.925054545197753426535918715729, −7.35230830142601247188023136304, −5.69146875848402258797863175685, −5.06898023368518580619222742805, −3.83652209701108997212525189487, −2.83311436785666374851377459104, −0.65566503017852838083783047216, 0.65566503017852838083783047216, 2.83311436785666374851377459104, 3.83652209701108997212525189487, 5.06898023368518580619222742805, 5.69146875848402258797863175685, 7.35230830142601247188023136304, 7.925054545197753426535918715729, 9.139696903754866133881869430916, 9.841003726026216146639662146285, 10.71219876121537935078554994302

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