Properties

Label 2-605-1.1-c1-0-26
Degree $2$
Conductor $605$
Sign $1$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.37·2-s + 3.26·3-s − 0.115·4-s + 5-s + 4.47·6-s − 1.37·7-s − 2.90·8-s + 7.63·9-s + 1.37·10-s − 0.377·12-s − 1.93·13-s − 1.88·14-s + 3.26·15-s − 3.75·16-s + 6.20·17-s + 10.4·18-s + 0.812·19-s − 0.115·20-s − 4.47·21-s − 3.63·23-s − 9.47·24-s + 25-s − 2.65·26-s + 15.1·27-s + 0.158·28-s − 7.83·29-s + 4.47·30-s + ⋯
L(s)  = 1  + 0.970·2-s + 1.88·3-s − 0.0578·4-s + 0.447·5-s + 1.82·6-s − 0.518·7-s − 1.02·8-s + 2.54·9-s + 0.434·10-s − 0.108·12-s − 0.535·13-s − 0.503·14-s + 0.842·15-s − 0.938·16-s + 1.50·17-s + 2.47·18-s + 0.186·19-s − 0.0258·20-s − 0.977·21-s − 0.758·23-s − 1.93·24-s + 0.200·25-s − 0.520·26-s + 2.91·27-s + 0.0300·28-s − 1.45·29-s + 0.817·30-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: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.826596465\)
\(L(\frac12)\) \(\approx\) \(3.826596465\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.37T + 2T^{2} \)
3 \( 1 - 3.26T + 3T^{2} \)
7 \( 1 + 1.37T + 7T^{2} \)
13 \( 1 + 1.93T + 13T^{2} \)
17 \( 1 - 6.20T + 17T^{2} \)
19 \( 1 - 0.812T + 19T^{2} \)
23 \( 1 + 3.63T + 23T^{2} \)
29 \( 1 + 7.83T + 29T^{2} \)
31 \( 1 + 3.40T + 31T^{2} \)
37 \( 1 + 4.52T + 37T^{2} \)
41 \( 1 - 1.82T + 41T^{2} \)
43 \( 1 + 6.46T + 43T^{2} \)
47 \( 1 - 4.73T + 47T^{2} \)
53 \( 1 + 8.39T + 53T^{2} \)
59 \( 1 - 1.11T + 59T^{2} \)
61 \( 1 + 2.54T + 61T^{2} \)
67 \( 1 - 2.73T + 67T^{2} \)
71 \( 1 + 1.11T + 71T^{2} \)
73 \( 1 + 12.4T + 73T^{2} \)
79 \( 1 - 5.80T + 79T^{2} \)
83 \( 1 - 15.9T + 83T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 - 14.1T + 97T^{2} \)
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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

−10.24883762600092535873327495815, −9.576766958184325668105283251077, −9.069500866227870833270336719473, −8.022531037116805957633296370858, −7.22653988411175688106777763196, −5.97272264687736990912616930854, −4.86854612007415761066388000758, −3.64954194556746814741035459736, −3.20079170385645795060093962102, −1.98505365307232056688200426516, 1.98505365307232056688200426516, 3.20079170385645795060093962102, 3.64954194556746814741035459736, 4.86854612007415761066388000758, 5.97272264687736990912616930854, 7.22653988411175688106777763196, 8.022531037116805957633296370858, 9.069500866227870833270336719473, 9.576766958184325668105283251077, 10.24883762600092535873327495815

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