↑

Properties

Label 2-2352-147.74-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.967 - 0.253i$
Analytic cond. $1.17380$
Root an. cond. $1.08342$
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  + (−0.365 − 0.930i)3-s + (−0.955 + 0.294i)7-s + (−0.733 + 0.680i)9-s + (−0.162 + 0.712i)13-s + (0.955 + 1.65i)19-s + (0.623 + 0.781i)21-s + (0.955 + 0.294i)25-s + (0.900 + 0.433i)27-s + (0.826 − 1.43i)31-s + (0.123 + 0.0841i)37-s + (0.722 − 0.108i)39-s + (0.914 + 1.14i)43-s + (0.826 − 0.563i)49-s + (1.19 − 1.49i)57-s + (−1.48 − 1.01i)61-s + ⋯
L(s)  = 1  + (−0.365 − 0.930i)3-s + (−0.955 + 0.294i)7-s + (−0.733 + 0.680i)9-s + (−0.162 + 0.712i)13-s + (0.955 + 1.65i)19-s + (0.623 + 0.781i)21-s + (0.955 + 0.294i)25-s + (0.900 + 0.433i)27-s + (0.826 − 1.43i)31-s + (0.123 + 0.0841i)37-s + (0.722 − 0.108i)39-s + (0.914 + 1.14i)43-s + (0.826 − 0.563i)49-s + (1.19 − 1.49i)57-s + (−1.48 − 1.01i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.253i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.967 - 0.253i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1985, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.967 - 0.253i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8515651725\)
\(L(\frac12)\) \(\approx\) \(0.8515651725\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.365 + 0.930i)T \)
7 \( 1 + (0.955 - 0.294i)T \)
good5 \( 1 + (-0.955 - 0.294i)T^{2} \)
11 \( 1 + (-0.0747 - 0.997i)T^{2} \)
13 \( 1 + (0.162 - 0.712i)T + (-0.900 - 0.433i)T^{2} \)
17 \( 1 + (0.988 + 0.149i)T^{2} \)
19 \( 1 + (-0.955 - 1.65i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.988 - 0.149i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 + (-0.826 + 1.43i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.123 - 0.0841i)T + (0.365 + 0.930i)T^{2} \)
41 \( 1 + (0.222 + 0.974i)T^{2} \)
43 \( 1 + (-0.914 - 1.14i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (-0.826 + 0.563i)T^{2} \)
53 \( 1 + (-0.365 + 0.930i)T^{2} \)
59 \( 1 + (-0.955 + 0.294i)T^{2} \)
61 \( 1 + (1.48 + 1.01i)T + (0.365 + 0.930i)T^{2} \)
67 \( 1 + (0.988 - 1.71i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.623 - 0.781i)T^{2} \)
73 \( 1 + (-0.142 - 0.0440i)T + (0.826 + 0.563i)T^{2} \)
79 \( 1 + (-0.826 - 1.43i)T + (-0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.0747 + 0.997i)T^{2} \)
97 \( 1 + 0.445T + T^{2} \)
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

−9.274566652044363601573198506789, −8.269211181907915376802335245124, −7.60795579555971577522740957214, −6.83274722055603322034693779651, −6.11110892211913164757002076792, −5.59395826260042051257825073994, −4.43849223342303982508019701109, −3.29136889473679109842857064712, −2.40267044638405866376129900135, −1.20157051551374921051934288539, 0.69787180881732726523699471557, 2.86056993384772504521251370914, 3.25405633585735294454910610242, 4.45524882377132853461996798135, 5.07807504145144399452514154016, 5.96255603054265350329879695987, 6.76157108140565712784460873309, 7.46610137526836241530453066162, 8.703316859001378075914378067122, 9.179033202391458932761822248449

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