Properties

Label 2-605-1.1-c1-0-23
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.09·2-s + 1.91·3-s + 2.39·4-s − 5-s − 4.00·6-s − 3.06·7-s − 0.817·8-s + 0.659·9-s + 2.09·10-s + 4.57·12-s + 3.04·13-s + 6.42·14-s − 1.91·15-s − 3.06·16-s − 0.463·17-s − 1.38·18-s − 7.89·19-s − 2.39·20-s − 5.86·21-s − 1.39·23-s − 1.56·24-s + 25-s − 6.39·26-s − 4.47·27-s − 7.33·28-s + 3.72·29-s + 4.00·30-s + ⋯
L(s)  = 1  − 1.48·2-s + 1.10·3-s + 1.19·4-s − 0.447·5-s − 1.63·6-s − 1.15·7-s − 0.289·8-s + 0.219·9-s + 0.662·10-s + 1.31·12-s + 0.845·13-s + 1.71·14-s − 0.493·15-s − 0.766·16-s − 0.112·17-s − 0.325·18-s − 1.81·19-s − 0.534·20-s − 1.28·21-s − 0.289·23-s − 0.319·24-s + 0.200·25-s − 1.25·26-s − 0.861·27-s − 1.38·28-s + 0.691·29-s + 0.731·30-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: \(1\)
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 \( 1 + 2.09T + 2T^{2} \)
3 \( 1 - 1.91T + 3T^{2} \)
7 \( 1 + 3.06T + 7T^{2} \)
13 \( 1 - 3.04T + 13T^{2} \)
17 \( 1 + 0.463T + 17T^{2} \)
19 \( 1 + 7.89T + 19T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 - 3.72T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 - 1.84T + 37T^{2} \)
41 \( 1 + 4.40T + 41T^{2} \)
43 \( 1 + 1.31T + 43T^{2} \)
47 \( 1 + 2.98T + 47T^{2} \)
53 \( 1 - 4.18T + 53T^{2} \)
59 \( 1 - 2.81T + 59T^{2} \)
61 \( 1 + 2.01T + 61T^{2} \)
67 \( 1 + 6.75T + 67T^{2} \)
71 \( 1 + 6.52T + 71T^{2} \)
73 \( 1 + 9.87T + 73T^{2} \)
79 \( 1 + 11.5T + 79T^{2} \)
83 \( 1 - 8.91T + 83T^{2} \)
89 \( 1 + 6.76T + 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−10.02448213931041315902410472382, −8.975479159007780974241480138803, −8.753240466125383673566522329912, −7.920841581654134894404328150083, −6.99950682901431182278685433213, −6.12026503765902627513151816615, −4.15454427297041553067919598302, −3.15263582214246796984365903508, −1.94021034518507580190273520563, 0, 1.94021034518507580190273520563, 3.15263582214246796984365903508, 4.15454427297041553067919598302, 6.12026503765902627513151816615, 6.99950682901431182278685433213, 7.920841581654134894404328150083, 8.753240466125383673566522329912, 8.975479159007780974241480138803, 10.02448213931041315902410472382

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