Properties

Label 2-2352-588.167-c0-0-0
Degree $2$
Conductor $2352$
Sign $-0.987 - 0.159i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.900 − 0.433i)7-s + (−0.900 + 0.433i)9-s + (−0.846 + 1.75i)13-s − 1.80·19-s + (0.222 − 0.974i)21-s + (0.900 − 0.433i)25-s + (−0.623 − 0.781i)27-s − 1.24·31-s + (−0.777 + 0.974i)37-s + (−1.90 − 0.433i)39-s + (0.846 + 0.193i)43-s + (0.623 + 0.781i)49-s + (−0.400 − 1.75i)57-s + (−1.22 − 0.974i)61-s + ⋯
L(s)  = 1  + (0.222 + 0.974i)3-s + (−0.900 − 0.433i)7-s + (−0.900 + 0.433i)9-s + (−0.846 + 1.75i)13-s − 1.80·19-s + (0.222 − 0.974i)21-s + (0.900 − 0.433i)25-s + (−0.623 − 0.781i)27-s − 1.24·31-s + (−0.777 + 0.974i)37-s + (−1.90 − 0.433i)39-s + (0.846 + 0.193i)43-s + (0.623 + 0.781i)49-s + (−0.400 − 1.75i)57-s + (−1.22 − 0.974i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.987 - 0.159i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1343, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5403813714\)
\(L(\frac12)\) \(\approx\) \(0.5403813714\)
\(L(1)\) 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 \( 1 + (-0.222 - 0.974i)T \)
7 \( 1 + (0.900 + 0.433i)T \)
good5 \( 1 + (-0.900 + 0.433i)T^{2} \)
11 \( 1 + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2} \)
17 \( 1 + (-0.222 - 0.974i)T^{2} \)
19 \( 1 + 1.80T + T^{2} \)
23 \( 1 + (-0.222 + 0.974i)T^{2} \)
29 \( 1 + (0.222 + 0.974i)T^{2} \)
31 \( 1 + 1.24T + T^{2} \)
37 \( 1 + (0.777 - 0.974i)T + (-0.222 - 0.974i)T^{2} \)
41 \( 1 + (-0.900 + 0.433i)T^{2} \)
43 \( 1 + (-0.846 - 0.193i)T + (0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 - 0.781i)T^{2} \)
53 \( 1 + (0.222 - 0.974i)T^{2} \)
59 \( 1 + (0.900 + 0.433i)T^{2} \)
61 \( 1 + (1.22 + 0.974i)T + (0.222 + 0.974i)T^{2} \)
67 \( 1 - 1.94iT - T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (-0.678 - 1.40i)T + (-0.623 + 0.781i)T^{2} \)
79 \( 1 - 1.56iT - T^{2} \)
83 \( 1 + (-0.623 + 0.781i)T^{2} \)
89 \( 1 + (0.623 - 0.781i)T^{2} \)
97 \( 1 + 0.867iT - T^{2} \)
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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

−9.461020715979829993653182569608, −8.983635679787533028317400812394, −8.230144390955337009129655463962, −6.99414485577534931986118863646, −6.60262387547900070002598268357, −5.53796184643977966561596725028, −4.41137760130188995991913336089, −4.14566040655343131505209510875, −3.00572595977367940992522778050, −2.06139177548181627442371401479, 0.32137833805072084174409341947, 2.02800284648896336199436471169, 2.84332239119603107924381601805, 3.61492148778531330229414223324, 5.05640584343658238903604878671, 5.83891433617304981860410593220, 6.48275693138313535966865563762, 7.34618593301765261759733308366, 7.900716488768856780534498607909, 8.881429499211967658479214454989

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