↑

Properties

Label 2-2352-7.2-c1-0-28
Degree $2$
Conductor $2352$
Sign $-0.605 + 0.795i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
Motivic weight $1$
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 − 4·13-s + (2 + 3.46i)17-s + (2 − 3.46i)19-s + (2 − 3.46i)23-s + (2.5 + 4.33i)25-s + 0.999·27-s + 2·29-s + (−4 − 6.92i)31-s + (3 − 5.19i)37-s + (2 + 3.46i)39-s − 12·41-s − 4·43-s + (4 − 6.92i)47-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.166 + 0.288i)9-s − 1.10·13-s + (0.485 + 0.840i)17-s + (0.458 − 0.794i)19-s + (0.417 − 0.722i)23-s + (0.5 + 0.866i)25-s + 0.192·27-s + 0.371·29-s + (−0.718 − 1.24i)31-s + (0.493 − 0.854i)37-s + (0.320 + 0.554i)39-s − 1.87·41-s − 0.609·43-s + (0.583 − 1.01i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.605 + 0.795i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9359983606\)
\(L(\frac12)\) \(\approx\) \(0.9359983606\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 12T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (-4 + 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-2 + 3.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (2 + 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (8 - 13.8i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + (-2 + 3.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 16T + 97T^{2} \)
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

−8.616805096405888598459051679579, −7.87856578404404832403684750817, −7.09004785642382670242818034884, −6.56436253318324356788327808373, −5.44542864399550228740508378506, −4.96444238012420921702311409628, −3.78783611334883933320798085046, −2.73430345961107663281163677563, −1.73864690582033924219784406854, −0.34515150874723563184372281334, 1.27430065273110689022248581241, 2.74408101028712344616133817415, 3.47940861977962158255908203966, 4.71351712165479301932974537044, 5.11902134504343211124055408588, 6.06361686954768365029160285067, 7.00887913678698432357808856323, 7.63547015041195256092559597679, 8.578320562584389877177774974543, 9.371035318737308712056444577341

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