Properties

Label 2-405-1.1-c1-0-15
Degree $2$
Conductor $405$
Sign $-1$
Analytic cond. $3.23394$
Root an. cond. $1.79831$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 4-s − 5-s − 3·7-s − 3·8-s − 10-s − 2·11-s − 2·13-s − 3·14-s − 16-s + 4·17-s − 8·19-s + 20-s − 2·22-s + 3·23-s + 25-s − 2·26-s + 3·28-s − 29-s + 5·32-s + 4·34-s + 3·35-s − 4·37-s − 8·38-s + 3·40-s + 5·41-s − 8·43-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s − 0.447·5-s − 1.13·7-s − 1.06·8-s − 0.316·10-s − 0.603·11-s − 0.554·13-s − 0.801·14-s − 1/4·16-s + 0.970·17-s − 1.83·19-s + 0.223·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s − 0.392·26-s + 0.566·28-s − 0.185·29-s + 0.883·32-s + 0.685·34-s + 0.507·35-s − 0.657·37-s − 1.29·38-s + 0.474·40-s + 0.780·41-s − 1.21·43-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: yes
Arithmetic: yes
Character: $\chi_{405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 405,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
7 \( 1 + 3 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 8 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 14 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 9 T + p T^{2} \)
89 \( 1 + 15 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−10.77469071826085469350446623927, −9.898527011341948118823413441919, −9.019286204430249539319170001575, −8.053933674867411547003077582597, −6.84595624283791199980948871072, −5.86971236166707955375003452449, −4.82687436247101660115281422977, −3.77402379572268668385012510927, −2.80534791056945823123328757931, 0, 2.80534791056945823123328757931, 3.77402379572268668385012510927, 4.82687436247101660115281422977, 5.86971236166707955375003452449, 6.84595624283791199980948871072, 8.053933674867411547003077582597, 9.019286204430249539319170001575, 9.898527011341948118823413441919, 10.77469071826085469350446623927

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