↑

Properties

Label 2-2352-7.4-c1-0-23
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 + (1 + 1.73i)5-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s − 2·13-s + 1.99·15-s + (3 − 5.19i)17-s + (2 + 3.46i)19-s + (0.500 − 0.866i)25-s − 0.999·27-s − 2·29-s + (−1.99 − 3.46i)33-s + (−3 − 5.19i)37-s + (−1 + 1.73i)39-s + 2·41-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s + (0.447 + 0.774i)5-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s − 0.554·13-s + 0.516·15-s + (0.727 − 1.26i)17-s + (0.458 + 0.794i)19-s + (0.100 − 0.173i)25-s − 0.192·27-s − 0.371·29-s + (−0.348 − 0.603i)33-s + (−0.493 − 0.854i)37-s + (−0.160 + 0.277i)39-s + 0.312·41-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: $\chi_{2352} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.605 + 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166858736\)
\(L(\frac12)\) \(\approx\) \(2.166858736\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 - 1.73i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2T + 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3 + 5.19i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
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 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (8 + 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18T + 97T^{2} \)
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

−8.986110837635124187554553854430, −7.912485992073285104694939895389, −7.39183450447668903519168005615, −6.54951996829193710377981229934, −5.90441797912309754295200006364, −5.07172018458693842342994873903, −3.69485898387662226937709894375, −3.02200297793579465275070352352, −2.10471627684087977317770597999, −0.78354883221851688463399060692, 1.25472338647344124511488483847, 2.24281320202841076040231901051, 3.45972651216623040377621903417, 4.36813759831279827187110938651, 5.04949071521281981685592426534, 5.79227885712163690276153584226, 6.85300057974494424073083374816, 7.60716961882873176513262596023, 8.525585438477936160443132809337, 9.141586222502462082609396679405

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