↑

Properties

Label 2-3620-3620.939-c0-0-1
Degree $2$
Conductor $3620$
Sign $0.920 + 0.390i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.997 − 0.0697i)2-s + (1.31 − 1.26i)3-s + (0.990 − 0.139i)4-s + (−0.104 + 0.994i)5-s + (1.22 − 1.35i)6-s + (−0.939 + 1.62i)7-s + (0.978 − 0.207i)8-s + (0.0816 − 2.33i)9-s + (−0.0348 + 0.999i)10-s + (1.12 − 1.43i)12-s + (−0.823 + 1.68i)14-s + (1.12 + 1.43i)15-s + (0.961 − 0.275i)16-s + (−0.0816 − 2.33i)18-s + (0.0348 + 0.999i)20-s + (0.830 + 3.33i)21-s + ⋯
L(s)  = 1  + (0.997 − 0.0697i)2-s + (1.31 − 1.26i)3-s + (0.990 − 0.139i)4-s + (−0.104 + 0.994i)5-s + (1.22 − 1.35i)6-s + (−0.939 + 1.62i)7-s + (0.978 − 0.207i)8-s + (0.0816 − 2.33i)9-s + (−0.0348 + 0.999i)10-s + (1.12 − 1.43i)12-s + (−0.823 + 1.68i)14-s + (1.12 + 1.43i)15-s + (0.961 − 0.275i)16-s + (−0.0816 − 2.33i)18-s + (0.0348 + 0.999i)20-s + (0.830 + 3.33i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3620\)    =    \(2^{2} \cdot 5 \cdot 181\)
Sign: $0.920 + 0.390i$
Analytic conductor: \(1.80661\)
Root analytic conductor: \(1.34410\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3620} (939, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3620,\ (\ :0),\ 0.920 + 0.390i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.443175439\)
\(L(\frac12)\) \(\approx\) \(3.443175439\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.997 + 0.0697i)T \)
5 \( 1 + (0.104 - 0.994i)T \)
181 \( 1 + (-0.990 + 0.139i)T \)
good3 \( 1 + (-1.31 + 1.26i)T + (0.0348 - 0.999i)T^{2} \)
7 \( 1 + (0.939 - 1.62i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.719 + 0.694i)T^{2} \)
13 \( 1 + (0.615 - 0.788i)T^{2} \)
17 \( 1 + (-0.766 - 0.642i)T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.22 + 0.762i)T + (0.438 - 0.898i)T^{2} \)
29 \( 1 + (-0.473 + 0.100i)T + (0.913 - 0.406i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.882 - 0.469i)T^{2} \)
41 \( 1 + (1.35 + 0.719i)T + (0.559 + 0.829i)T^{2} \)
43 \( 1 + (0.194 - 1.10i)T + (-0.939 - 0.342i)T^{2} \)
47 \( 1 + (1.20 - 1.54i)T + (-0.241 - 0.970i)T^{2} \)
53 \( 1 + (-0.559 + 0.829i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.35 + 1.13i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.139 + 0.155i)T + (-0.104 - 0.994i)T^{2} \)
71 \( 1 + (-0.669 - 0.743i)T^{2} \)
73 \( 1 + (-0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.559 + 0.829i)T^{2} \)
83 \( 1 + (0.345 + 0.512i)T + (-0.374 + 0.927i)T^{2} \)
89 \( 1 + (-1.51 + 1.27i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.719 - 0.694i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.514699254963674409759519389104, −7.79896054905147050745732605521, −7.01462437866674469341157810506, −6.37863542205680953995177905881, −6.14485274021310380403226072173, −4.89448415933068152806579077832, −3.46915471714906855449650169907, −3.00976571258957344197381807783, −2.55644616569526024142254613020, −1.72048285867254289771098786150, 1.49684660252552202063979051502, 2.88935064293251140378519912403, 3.57234381519169367984301013591, 3.99030529857433174294986630776, 4.77802269258256443286399397753, 5.28738423679034701020610850372, 6.62485855000421499448044597513, 7.33740900235978505347535107543, 8.038199372641416430626970499931, 8.778266216799984473646385638044

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