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Properties

Label 2-2400-600.221-c0-0-1
Degree $2$
Conductor $2400$
Sign $0.968 - 0.248i$
Analytic cond. $1.19775$
Root an. cond. $1.09442$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + 1.61·7-s + (0.309 + 0.951i)9-s + (−0.190 + 0.587i)11-s + (0.809 − 0.587i)15-s + (1.30 + 0.951i)21-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−1.61 − 1.17i)29-s + (−1.30 + 0.951i)31-s + (−0.5 + 0.363i)33-s + (0.500 − 1.53i)35-s + 45-s + 1.61·49-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + 1.61·7-s + (0.309 + 0.951i)9-s + (−0.190 + 0.587i)11-s + (0.809 − 0.587i)15-s + (1.30 + 0.951i)21-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (−1.61 − 1.17i)29-s + (−1.30 + 0.951i)31-s + (−0.5 + 0.363i)33-s + (0.500 − 1.53i)35-s + 45-s + 1.61·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.248i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2400\)    =    \(2^{5} \cdot 3 \cdot 5^{2}\)
Sign: $0.968 - 0.248i$
Analytic conductor: \(1.19775\)
Root analytic conductor: \(1.09442\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2400} (2321, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2400,\ (\ :0),\ 0.968 - 0.248i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.912792142\)
\(L(\frac12)\) \(\approx\) \(1.912792142\)
\(L(1)\) 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 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.309 + 0.951i)T \)
good7 \( 1 - 1.61T + T^{2} \)
11 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (0.809 - 0.587i)T^{2} \)
17 \( 1 + (-0.309 + 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.809 + 0.587i)T^{2} \)
29 \( 1 + (1.61 + 1.17i)T + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (0.809 - 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (-0.309 + 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (-0.618 + 1.90i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.30 - 0.951i)T + (0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.213371897988672045169944797548, −8.365991633124262822009147566847, −7.916269797112415620180980767860, −7.22249536025577522215955185203, −5.70385901134460923806747863395, −5.03652139829616388210572762846, −4.50068941060896127828266886315, −3.65717623081512764432349883103, −2.15822935580019871561212320632, −1.64725485423894307757437029840, 1.53487797466203927128887120817, 2.23265110314282076372676184625, 3.26200224699653132813398785879, 4.09684728435255751496246669154, 5.33835509817559802238086479928, 6.01308321894374197007384544654, 7.16458879115362193675156752505, 7.50702690950234684907248825611, 8.269112594541201560355375242119, 8.972270378433361358479358227957

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