Properties

Label 2-185150-1.1-c1-0-3
Degree $2$
Conductor $185150$
Sign $1$
Analytic cond. $1478.43$
Root an. cond. $38.4503$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s − 7-s − 8-s − 3·9-s − 4·11-s + 3·13-s + 14-s + 16-s − 17-s + 3·18-s + 4·22-s − 3·26-s − 28-s − 4·31-s − 32-s + 34-s − 3·36-s + 11·37-s − 10·41-s − 2·43-s − 4·44-s − 11·47-s + 49-s + 3·52-s + 53-s + 56-s + ⋯
L(s)  = 1  − 0.707·2-s + 1/2·4-s − 0.377·7-s − 0.353·8-s − 9-s − 1.20·11-s + 0.832·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 0.707·18-s + 0.852·22-s − 0.588·26-s − 0.188·28-s − 0.718·31-s − 0.176·32-s + 0.171·34-s − 1/2·36-s + 1.80·37-s − 1.56·41-s − 0.304·43-s − 0.603·44-s − 1.60·47-s + 1/7·49-s + 0.416·52-s + 0.137·53-s + 0.133·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(185150\)    =    \(2 \cdot 5^{2} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1478.43\)
Root analytic conductor: \(38.4503\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 185150,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3651020642\)
\(L(\frac12)\) \(\approx\) \(0.3651020642\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
7 \( 1 + T \)
23 \( 1 \)
good3 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 11 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 4 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 11 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 + T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−13.22815880795123, −12.66590012650624, −12.15979326549884, −11.47024835275461, −11.20968632573969, −10.86213435075897, −10.22359561793771, −9.858214598886502, −9.327280425453166, −8.795543400651866, −8.332330387027251, −8.016302625998569, −7.541313182967791, −6.781167572657694, −6.451927441122649, −5.838749970110652, −5.445439787661608, −4.888678561358026, −4.160205002805248, −3.338664780565563, −3.098469804766069, −2.421272133217831, −1.869021430826074, −1.054182208219933, −0.2099348975888549, 0.2099348975888549, 1.054182208219933, 1.869021430826074, 2.421272133217831, 3.098469804766069, 3.338664780565563, 4.160205002805248, 4.888678561358026, 5.445439787661608, 5.838749970110652, 6.451927441122649, 6.781167572657694, 7.541313182967791, 8.016302625998569, 8.332330387027251, 8.795543400651866, 9.327280425453166, 9.858214598886502, 10.22359561793771, 10.86213435075897, 11.20968632573969, 11.47024835275461, 12.15979326549884, 12.66590012650624, 13.22815880795123

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