Properties

Label 2-605-1.1-c1-0-22
Degree $2$
Conductor $605$
Sign $-1$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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 − 3·3-s − 4-s + 5-s − 3·6-s + 3·7-s − 3·8-s + 6·9-s + 10-s + 3·12-s − 4·13-s + 3·14-s − 3·15-s − 16-s + 6·18-s − 4·19-s − 20-s − 9·21-s − 8·23-s + 9·24-s + 25-s − 4·26-s − 9·27-s − 3·28-s − 6·29-s − 3·30-s − 2·31-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s − 1/2·4-s + 0.447·5-s − 1.22·6-s + 1.13·7-s − 1.06·8-s + 2·9-s + 0.316·10-s + 0.866·12-s − 1.10·13-s + 0.801·14-s − 0.774·15-s − 1/4·16-s + 1.41·18-s − 0.917·19-s − 0.223·20-s − 1.96·21-s − 1.66·23-s + 1.83·24-s + 1/5·25-s − 0.784·26-s − 1.73·27-s − 0.566·28-s − 1.11·29-s − 0.547·30-s − 0.359·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{605} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -1)\)

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)$
bad5 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
3 \( 1 + p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 4 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 11 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−10.38032562538173978810669186344, −9.664566365355955271051110792478, −8.423723216909326273117880530024, −7.25464453677185570179698479482, −6.15837200774450811108804773127, −5.45422582252650267450469418523, −4.84024310064756519754897603006, −4.05163886242755682285675106710, −1.91615236151570534807396468396, 0, 1.91615236151570534807396468396, 4.05163886242755682285675106710, 4.84024310064756519754897603006, 5.45422582252650267450469418523, 6.15837200774450811108804773127, 7.25464453677185570179698479482, 8.423723216909326273117880530024, 9.664566365355955271051110792478, 10.38032562538173978810669186344

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