↑

Properties

Label 2-2352-84.59-c0-0-1
Degree $2$
Conductor $2352$
Sign $0.832 + 0.553i$
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  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s − 1.73i·13-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (1.49 + 0.866i)39-s + 1.73i·43-s + 0.999·57-s + (−1.5 − 0.866i)67-s + (1.5 + 0.866i)73-s + (0.499 + 0.866i)75-s + (1.5 − 0.866i)79-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)3-s + (−0.499 − 0.866i)9-s − 1.73i·13-s + (−0.5 − 0.866i)19-s + (0.5 − 0.866i)25-s + 0.999·27-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)37-s + (1.49 + 0.866i)39-s + 1.73i·43-s + 0.999·57-s + (−1.5 − 0.866i)67-s + (1.5 + 0.866i)73-s + (0.499 + 0.866i)75-s + (1.5 − 0.866i)79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.832 + 0.553i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.832 + 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8551471383\)
\(L(\frac12)\) \(\approx\) \(0.8551471383\)
\(L(1)\) 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 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73iT - T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−9.208402561784249129403354128363, −8.387974152592066553983314740843, −7.68773764055467617950474497285, −6.54970445693668472833477945798, −5.91883268400493973093050510489, −5.07767896348740183805831303840, −4.42305005267351053565529601173, −3.38146687422328634116103421430, −2.57378002851777410738913291666, −0.65229264616749066761673331274, 1.41000418042886420178066325177, 2.20767064479174365444230827841, 3.51833932078734243804777823229, 4.58410708430356239827825179234, 5.37539195037049026172575645262, 6.35261099496016856862426474389, 6.83030911364757468548076554435, 7.54690697282397223345786221013, 8.507345367718888280854755979823, 9.027932805887989842587486150410

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