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Properties

Label 2-2352-588.467-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  + (0.955 − 0.294i)3-s + (−0.0747 + 0.997i)7-s + (0.826 − 0.563i)9-s + (0.255 + 0.531i)13-s + (0.0747 − 0.129i)19-s + (0.222 + 0.974i)21-s + (−0.0747 − 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (0.722 − 0.108i)37-s + (0.400 + 0.432i)39-s + (−1.09 + 0.250i)43-s + (−0.988 − 0.149i)49-s + (0.0332 − 0.145i)57-s + (−0.233 − 1.54i)61-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)3-s + (−0.0747 + 0.997i)7-s + (0.826 − 0.563i)9-s + (0.255 + 0.531i)13-s + (0.0747 − 0.129i)19-s + (0.222 + 0.974i)21-s + (−0.0747 − 0.997i)25-s + (0.623 − 0.781i)27-s + (0.988 + 1.71i)31-s + (0.722 − 0.108i)37-s + (0.400 + 0.432i)39-s + (−1.09 + 0.250i)43-s + (−0.988 − 0.149i)49-s + (0.0332 − 0.145i)57-s + (−0.233 − 1.54i)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: $\chi_{2352} (1055, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.987 - 0.159i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.706967864\)
\(L(\frac12)\) \(\approx\) \(1.706967864\)
\(L(1)\) 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 \( 1 + (-0.955 + 0.294i)T \)
7 \( 1 + (0.0747 - 0.997i)T \)
good5 \( 1 + (0.0747 + 0.997i)T^{2} \)
11 \( 1 + (0.365 + 0.930i)T^{2} \)
13 \( 1 + (-0.255 - 0.531i)T + (-0.623 + 0.781i)T^{2} \)
17 \( 1 + (-0.733 - 0.680i)T^{2} \)
19 \( 1 + (-0.0747 + 0.129i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.733 + 0.680i)T^{2} \)
29 \( 1 + (0.222 - 0.974i)T^{2} \)
31 \( 1 + (-0.988 - 1.71i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.722 + 0.108i)T + (0.955 - 0.294i)T^{2} \)
41 \( 1 + (-0.900 - 0.433i)T^{2} \)
43 \( 1 + (1.09 - 0.250i)T + (0.900 - 0.433i)T^{2} \)
47 \( 1 + (0.988 + 0.149i)T^{2} \)
53 \( 1 + (-0.955 - 0.294i)T^{2} \)
59 \( 1 + (-0.0747 + 0.997i)T^{2} \)
61 \( 1 + (0.233 + 1.54i)T + (-0.955 + 0.294i)T^{2} \)
67 \( 1 + (1.17 - 0.680i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.222 - 0.974i)T^{2} \)
73 \( 1 + (-1.85 + 0.139i)T + (0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.258 + 0.149i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.623 - 0.781i)T^{2} \)
89 \( 1 + (0.365 - 0.930i)T^{2} \)
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

−9.085178894855516449239439719141, −8.412901428950399406448322205441, −7.925847958964666486851844507077, −6.75088530146806863212635657622, −6.36907379998295425058051536136, −5.17847506473919038882922187712, −4.30392902700946112367667590703, −3.23566745266946186648757365311, −2.50973342558212575207553800728, −1.50637813220103621201778098390, 1.25300934939407960684611742031, 2.53848156751059836962600290622, 3.48576569688225305305302778220, 4.13806038553738945329885435615, 5.00108465372538484471294723053, 6.10499175913519489503932452085, 7.05899852314734237604968973333, 7.76921043674197711556366848522, 8.240068068634462539837126180966, 9.244596866226634031895183929949

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