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Properties

Label 2-7605-1.1-c1-0-253
Degree $2$
Conductor $7605$
Sign $-1$
Analytic cond. $60.7262$
Root an. cond. $7.79270$
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  + 2·2-s + 2·4-s + 5-s + 3·7-s + 2·10-s − 5·11-s + 6·14-s − 4·16-s − 5·17-s − 2·19-s + 2·20-s − 10·22-s + 23-s + 25-s + 6·28-s − 10·29-s + 2·31-s − 8·32-s − 10·34-s + 3·35-s + 3·37-s − 4·38-s − 9·41-s − 4·43-s − 10·44-s + 2·46-s + 10·47-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.447·5-s + 1.13·7-s + 0.632·10-s − 1.50·11-s + 1.60·14-s − 16-s − 1.21·17-s − 0.458·19-s + 0.447·20-s − 2.13·22-s + 0.208·23-s + 1/5·25-s + 1.13·28-s − 1.85·29-s + 0.359·31-s − 1.41·32-s − 1.71·34-s + 0.507·35-s + 0.493·37-s − 0.648·38-s − 1.40·41-s − 0.609·43-s − 1.50·44-s + 0.294·46-s + 1.45·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7605\)    =    \(3^{2} \cdot 5 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(60.7262\)
Root analytic conductor: \(7.79270\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 7605,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 9 T + p T^{2} \)
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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

−7.40547395663770579012954332221, −6.62041415913029493471167225435, −5.86475683381808899692173764454, −5.24227176253115492205485345417, −4.80019935028895801751220520872, −4.17042727719485350984478313679, −3.19276093716699234142540547398, −2.35315445979924710503734826901, −1.81034027339283215405987905059, 0, 1.81034027339283215405987905059, 2.35315445979924710503734826901, 3.19276093716699234142540547398, 4.17042727719485350984478313679, 4.80019935028895801751220520872, 5.24227176253115492205485345417, 5.86475683381808899692173764454, 6.62041415913029493471167225435, 7.40547395663770579012954332221

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