Properties

Label 2-41405-1.1-c1-0-3
Degree $2$
Conductor $41405$
Sign $1$
Analytic cond. $330.620$
Root an. cond. $18.1829$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s − 2·9-s + 3·11-s + 2·12-s + 15-s + 4·16-s − 3·17-s + 2·19-s + 2·20-s − 6·23-s + 25-s + 5·27-s + 3·29-s − 4·31-s − 3·33-s + 4·36-s − 2·37-s − 12·41-s − 10·43-s − 6·44-s + 2·45-s + 9·47-s − 4·48-s + 3·51-s + 12·53-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s + 0.904·11-s + 0.577·12-s + 0.258·15-s + 16-s − 0.727·17-s + 0.458·19-s + 0.447·20-s − 1.25·23-s + 1/5·25-s + 0.962·27-s + 0.557·29-s − 0.718·31-s − 0.522·33-s + 2/3·36-s − 0.328·37-s − 1.87·41-s − 1.52·43-s − 0.904·44-s + 0.298·45-s + 1.31·47-s − 0.577·48-s + 0.420·51-s + 1.64·53-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 41405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(41405\)    =    \(5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(330.620\)
Root analytic conductor: \(18.1829\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{41405} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 41405,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4623028957\)
\(L(\frac12)\) \(\approx\) \(0.4623028957\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + T \)
7 \( 1 \)
13 \( 1 \)
good2 \( 1 + p T^{2} \)
3 \( 1 + T + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 12 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−14.73745345006956, −14.09541738144420, −13.73201683413446, −13.37560523200741, −12.43791849004963, −12.14829674417630, −11.70015344582317, −11.23970877266190, −10.43040736530272, −10.14004666497211, −9.373345651997256, −8.842824832479709, −8.476137813606813, −7.961529784856642, −7.128763540209931, −6.603637092018811, −5.984287675883349, −5.364246026512435, −4.910289088720229, −4.189939124829229, −3.716174145372085, −3.111054668398681, −2.079784218480817, −1.181238802550996, −0.2888610215191289, 0.2888610215191289, 1.181238802550996, 2.079784218480817, 3.111054668398681, 3.716174145372085, 4.189939124829229, 4.910289088720229, 5.364246026512435, 5.984287675883349, 6.603637092018811, 7.128763540209931, 7.961529784856642, 8.476137813606813, 8.842824832479709, 9.373345651997256, 10.14004666497211, 10.43040736530272, 11.23970877266190, 11.70015344582317, 12.14829674417630, 12.43791849004963, 13.37560523200741, 13.73201683413446, 14.09541738144420, 14.73745345006956

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