Properties

Label 2-605-1.1-c1-0-21
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 − 4-s + 5-s + 3·8-s − 3·9-s − 10-s − 2·13-s − 16-s − 6·17-s + 3·18-s + 4·19-s − 20-s + 4·23-s + 25-s + 2·26-s − 6·29-s − 8·31-s − 5·32-s + 6·34-s + 3·36-s − 2·37-s − 4·38-s + 3·40-s − 2·41-s − 4·43-s − 3·45-s − 4·46-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 0.447·5-s + 1.06·8-s − 9-s − 0.316·10-s − 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.917·19-s − 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.392·26-s − 1.11·29-s − 1.43·31-s − 0.883·32-s + 1.02·34-s + 1/2·36-s − 0.328·37-s − 0.648·38-s + 0.474·40-s − 0.312·41-s − 0.609·43-s − 0.447·45-s − 0.589·46-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: Trivial
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 \( 1 + p T^{2} \)
7 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 10 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−10.02041446476298683486486191886, −9.196982768972178400591968116867, −8.758832935924856173060099057787, −7.71367077872044547263533331174, −6.79405906039010685182823940446, −5.50821445395345299948641768104, −4.77278357968932846404359363084, −3.32452113574721608186627592149, −1.86853886263346253357248352832, 0, 1.86853886263346253357248352832, 3.32452113574721608186627592149, 4.77278357968932846404359363084, 5.50821445395345299948641768104, 6.79405906039010685182823940446, 7.71367077872044547263533331174, 8.758832935924856173060099057787, 9.196982768972178400591968116867, 10.02041446476298683486486191886

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