Properties

Label 2-405-1.1-c1-0-10
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

−10.32043257877866718022672062677, −9.946177052534403132870451803520, −8.964305577132556078601253128051, −8.406907006225460062267410041104, −6.95907981187563203473208917697, −6.29557968587596560657178728455, −4.84628635731234228377894729907, −3.75836640301739809806939141128, −2.06014522909482997436709887093, 0, 2.06014522909482997436709887093, 3.75836640301739809806939141128, 4.84628635731234228377894729907, 6.29557968587596560657178728455, 6.95907981187563203473208917697, 8.406907006225460062267410041104, 8.964305577132556078601253128051, 9.946177052534403132870451803520, 10.32043257877866718022672062677

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