Properties

Label 2-605-1.1-c1-0-7
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  − 1.87·2-s − 1.77·3-s + 1.51·4-s + 5-s + 3.33·6-s + 4.25·7-s + 0.901·8-s + 0.159·9-s − 1.87·10-s − 2.70·12-s + 1.36·13-s − 7.98·14-s − 1.77·15-s − 4.73·16-s + 2.09·17-s − 0.299·18-s + 0.604·19-s + 1.51·20-s − 7.56·21-s − 4.39·23-s − 1.60·24-s + 25-s − 2.55·26-s + 5.04·27-s + 6.46·28-s − 6.63·29-s + 3.33·30-s + ⋯
L(s)  = 1  − 1.32·2-s − 1.02·3-s + 0.759·4-s + 0.447·5-s + 1.36·6-s + 1.60·7-s + 0.318·8-s + 0.0531·9-s − 0.593·10-s − 0.779·12-s + 0.377·13-s − 2.13·14-s − 0.458·15-s − 1.18·16-s + 0.508·17-s − 0.0705·18-s + 0.138·19-s + 0.339·20-s − 1.65·21-s − 0.916·23-s − 0.327·24-s + 0.200·25-s − 0.500·26-s + 0.971·27-s + 1.22·28-s − 1.23·29-s + 0.608·30-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

\(L(1)\) \(\approx\) \(0.6447420072\)
\(L(\frac12)\) \(\approx\) \(0.6447420072\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.87T + 2T^{2} \)
3 \( 1 + 1.77T + 3T^{2} \)
7 \( 1 - 4.25T + 7T^{2} \)
13 \( 1 - 1.36T + 13T^{2} \)
17 \( 1 - 2.09T + 17T^{2} \)
19 \( 1 - 0.604T + 19T^{2} \)
23 \( 1 + 4.39T + 23T^{2} \)
29 \( 1 + 6.63T + 29T^{2} \)
31 \( 1 + 2.19T + 31T^{2} \)
37 \( 1 - 6.16T + 37T^{2} \)
41 \( 1 - 7.40T + 41T^{2} \)
43 \( 1 - 12.6T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 + 6.65T + 53T^{2} \)
59 \( 1 - 12.0T + 59T^{2} \)
61 \( 1 + 5.68T + 61T^{2} \)
67 \( 1 - 9.86T + 67T^{2} \)
71 \( 1 + 5.23T + 71T^{2} \)
73 \( 1 + 0.722T + 73T^{2} \)
79 \( 1 - 5.67T + 79T^{2} \)
83 \( 1 + 0.952T + 83T^{2} \)
89 \( 1 - 1.24T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−10.81923980747304794773415624293, −9.789450759579999885227731640820, −8.941778165051189525738264100272, −8.030009234004533529306819749843, −7.44807360774094599999828725945, −6.10107983838590632936608903311, −5.33322061232929930404373761811, −4.30527704672202159909961329509, −2.08303021296616042965844033710, −0.948431305586601174813472817241, 0.948431305586601174813472817241, 2.08303021296616042965844033710, 4.30527704672202159909961329509, 5.33322061232929930404373761811, 6.10107983838590632936608903311, 7.44807360774094599999828725945, 8.030009234004533529306819749843, 8.941778165051189525738264100272, 9.789450759579999885227731640820, 10.81923980747304794773415624293

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