Properties

Label 2-405-1.1-c3-0-4
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  + 0.225·2-s − 7.94·4-s + 5·5-s − 31.1·7-s − 3.59·8-s + 1.12·10-s − 18.1·11-s − 50.1·13-s − 7.02·14-s + 62.7·16-s + 131.·17-s + 23.2·19-s − 39.7·20-s − 4.08·22-s + 32.9·23-s + 25·25-s − 11.2·26-s + 247.·28-s − 125.·29-s + 125.·31-s + 42.8·32-s + 29.6·34-s − 155.·35-s + 99.9·37-s + 5.23·38-s − 17.9·40-s − 245.·41-s + ⋯
L(s)  = 1  + 0.0796·2-s − 0.993·4-s + 0.447·5-s − 1.68·7-s − 0.158·8-s + 0.0356·10-s − 0.496·11-s − 1.07·13-s − 0.134·14-s + 0.981·16-s + 1.87·17-s + 0.280·19-s − 0.444·20-s − 0.0395·22-s + 0.299·23-s + 0.200·25-s − 0.0852·26-s + 1.67·28-s − 0.805·29-s + 0.724·31-s + 0.236·32-s + 0.149·34-s − 0.753·35-s + 0.444·37-s + 0.0223·38-s − 0.0710·40-s − 0.934·41-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

\(L(2)\) \(\approx\) \(1.058538194\)
\(L(\frac12)\) \(\approx\) \(1.058538194\)
\(L(\frac{5}{2})\) 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 \( 1 - 0.225T + 8T^{2} \)
7 \( 1 + 31.1T + 343T^{2} \)
11 \( 1 + 18.1T + 1.33e3T^{2} \)
13 \( 1 + 50.1T + 2.19e3T^{2} \)
17 \( 1 - 131.T + 4.91e3T^{2} \)
19 \( 1 - 23.2T + 6.85e3T^{2} \)
23 \( 1 - 32.9T + 1.21e4T^{2} \)
29 \( 1 + 125.T + 2.43e4T^{2} \)
31 \( 1 - 125.T + 2.97e4T^{2} \)
37 \( 1 - 99.9T + 5.06e4T^{2} \)
41 \( 1 + 245.T + 6.89e4T^{2} \)
43 \( 1 - 139.T + 7.95e4T^{2} \)
47 \( 1 - 472.T + 1.03e5T^{2} \)
53 \( 1 + 421.T + 1.48e5T^{2} \)
59 \( 1 - 742.T + 2.05e5T^{2} \)
61 \( 1 - 8.97T + 2.26e5T^{2} \)
67 \( 1 - 588.T + 3.00e5T^{2} \)
71 \( 1 - 48.5T + 3.57e5T^{2} \)
73 \( 1 - 409.T + 3.89e5T^{2} \)
79 \( 1 - 530.T + 4.93e5T^{2} \)
83 \( 1 - 294.T + 5.71e5T^{2} \)
89 \( 1 + 852.T + 7.04e5T^{2} \)
97 \( 1 + 388.T + 9.12e5T^{2} \)
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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

−10.41848777330745995747250579784, −9.718574894443736854808226081763, −9.426404286835289086574115529521, −8.093104052849926694559316666476, −7.08162077865792301350826944255, −5.84955259337724893680988145165, −5.15546169278778745004258186009, −3.70795399960364442790559628130, −2.79306455443702338177852482022, −0.65008926217116654458751986923, 0.65008926217116654458751986923, 2.79306455443702338177852482022, 3.70795399960364442790559628130, 5.15546169278778745004258186009, 5.84955259337724893680988145165, 7.08162077865792301350826944255, 8.093104052849926694559316666476, 9.426404286835289086574115529521, 9.718574894443736854808226081763, 10.41848777330745995747250579784

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