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Properties

Label 2-3620-3620.2479-c0-0-0
Degree $2$
Conductor $3620$
Sign $0.994 - 0.104i$
Analytic cond. $1.80661$
Root an. cond. $1.34410$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.615 − 0.788i)2-s + (−1.13 + 0.709i)3-s + (−0.241 − 0.970i)4-s + (−0.978 + 0.207i)5-s + (−0.139 + 1.33i)6-s + (0.173 − 0.300i)7-s + (−0.913 − 0.406i)8-s + (0.346 − 0.710i)9-s + (−0.438 + 0.898i)10-s + (0.962 + 0.929i)12-s + (−0.130 − 0.322i)14-s + (0.962 − 0.929i)15-s + (−0.882 + 0.469i)16-s + (−0.346 − 0.710i)18-s + (0.438 + 0.898i)20-s + (0.0162 + 0.464i)21-s + ⋯
L(s)  = 1  + (0.615 − 0.788i)2-s + (−1.13 + 0.709i)3-s + (−0.241 − 0.970i)4-s + (−0.978 + 0.207i)5-s + (−0.139 + 1.33i)6-s + (0.173 − 0.300i)7-s + (−0.913 − 0.406i)8-s + (0.346 − 0.710i)9-s + (−0.438 + 0.898i)10-s + (0.962 + 0.929i)12-s + (−0.130 − 0.322i)14-s + (0.962 − 0.929i)15-s + (−0.882 + 0.469i)16-s + (−0.346 − 0.710i)18-s + (0.438 + 0.898i)20-s + (0.0162 + 0.464i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $0.994 - 0.104i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (2479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ 0.994 - 0.104i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7579341377\)
\(L(\frac12)\) \(\approx\) \(0.7579341377\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.615 + 0.788i)T \)
5 \( 1 + (0.978 - 0.207i)T \)
181 \( 1 + (0.241 + 0.970i)T \)
good3 \( 1 + (1.13 - 0.709i)T + (0.438 - 0.898i)T^{2} \)
7 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.848 - 0.529i)T^{2} \)
13 \( 1 + (0.719 + 0.694i)T^{2} \)
17 \( 1 + (0.939 - 0.342i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (0.948 - 1.40i)T + (-0.374 - 0.927i)T^{2} \)
29 \( 1 + (-0.0637 - 0.0283i)T + (0.669 + 0.743i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (0.997 - 0.0697i)T^{2} \)
41 \( 1 + (-1.87 - 0.131i)T + (0.990 + 0.139i)T^{2} \)
43 \( 1 + (1.51 + 1.27i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-1.31 - 1.26i)T + (0.0348 + 0.999i)T^{2} \)
53 \( 1 + (-0.990 + 0.139i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-1.87 + 0.682i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.204 - 1.94i)T + (-0.978 - 0.207i)T^{2} \)
71 \( 1 + (0.104 + 0.994i)T^{2} \)
73 \( 1 + (-0.766 + 0.642i)T^{2} \)
79 \( 1 + (-0.990 + 0.139i)T^{2} \)
83 \( 1 + (-1.60 - 0.225i)T + (0.961 + 0.275i)T^{2} \)
89 \( 1 + (-0.454 - 0.165i)T + (0.766 + 0.642i)T^{2} \)
97 \( 1 + (-0.848 + 0.529i)T^{2} \)
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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

−8.972861668545247753931569447918, −7.936396872767122801555102869782, −7.14031809310361656796192428440, −6.17593846690528615038475294675, −5.53989450688940719211976147465, −4.81285168494529150375142153658, −4.06182118000948317885822359320, −3.64110526336217330288119916340, −2.40924430722174157090223257668, −0.918708451955647259711458770550, 0.56025392672855759207227583780, 2.28211083719480583687551452253, 3.51318362981348121500673838376, 4.37574135301417024477305696259, 5.03987916023121179095204123800, 5.81695302217413137278842762284, 6.45581524868144995518168184980, 7.07537133211084693259722092975, 7.78392991143641083032271831066, 8.393411357447625662706315173298

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