Properties

Label 2-2352-588.83-c0-0-0
Degree $2$
Conductor $2352$
Sign $-0.253 - 0.967i$
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  + (−0.623 + 0.781i)3-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)9-s + (1.52 + 0.347i)13-s − 0.445·19-s + (−0.623 − 0.781i)21-s + (0.222 + 0.974i)25-s + (0.900 + 0.433i)27-s + 1.80·31-s + (−1.62 + 0.781i)37-s + (−1.22 + 0.974i)39-s + (−1.52 + 1.21i)43-s + (−0.900 − 0.433i)49-s + (0.277 − 0.347i)57-s + (−0.376 − 0.781i)61-s + ⋯
L(s)  = 1  + (−0.623 + 0.781i)3-s + (−0.222 + 0.974i)7-s + (−0.222 − 0.974i)9-s + (1.52 + 0.347i)13-s − 0.445·19-s + (−0.623 − 0.781i)21-s + (0.222 + 0.974i)25-s + (0.900 + 0.433i)27-s + 1.80·31-s + (−1.62 + 0.781i)37-s + (−1.22 + 0.974i)39-s + (−1.52 + 1.21i)43-s + (−0.900 − 0.433i)49-s + (0.277 − 0.347i)57-s + (−0.376 − 0.781i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.253 - 0.967i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (671, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ -0.253 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8998503948\)
\(L(\frac12)\) \(\approx\) \(0.8998503948\)
\(L(1)\) 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 \( 1 + (0.623 - 0.781i)T \)
7 \( 1 + (0.222 - 0.974i)T \)
good5 \( 1 + (-0.222 - 0.974i)T^{2} \)
11 \( 1 + (-0.900 - 0.433i)T^{2} \)
13 \( 1 + (-1.52 - 0.347i)T + (0.900 + 0.433i)T^{2} \)
17 \( 1 + (0.623 - 0.781i)T^{2} \)
19 \( 1 + 0.445T + T^{2} \)
23 \( 1 + (0.623 + 0.781i)T^{2} \)
29 \( 1 + (-0.623 + 0.781i)T^{2} \)
31 \( 1 - 1.80T + T^{2} \)
37 \( 1 + (1.62 - 0.781i)T + (0.623 - 0.781i)T^{2} \)
41 \( 1 + (-0.222 - 0.974i)T^{2} \)
43 \( 1 + (1.52 - 1.21i)T + (0.222 - 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (-0.623 - 0.781i)T^{2} \)
59 \( 1 + (0.222 - 0.974i)T^{2} \)
61 \( 1 + (0.376 + 0.781i)T + (-0.623 + 0.781i)T^{2} \)
67 \( 1 - 1.56iT - T^{2} \)
71 \( 1 + (0.623 + 0.781i)T^{2} \)
73 \( 1 + (-0.846 + 0.193i)T + (0.900 - 0.433i)T^{2} \)
79 \( 1 + 0.867iT - T^{2} \)
83 \( 1 + (0.900 - 0.433i)T^{2} \)
89 \( 1 + (-0.900 + 0.433i)T^{2} \)
97 \( 1 - 1.94iT - T^{2} \)
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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

−9.366176225525895904858723636490, −8.716646217121041472310414546473, −8.175030017413761137175756752368, −6.61496354846829600230477167721, −6.35873092574648860767359434058, −5.42332752677594433292090150018, −4.74288479931963217122658685460, −3.70673848974345101078448718270, −2.97727333996728002763040828790, −1.46523429046331886913914401604, 0.71995210668647087997647662698, 1.82493380605408697258664078838, 3.17488670903551823327707411186, 4.12989585224044429505395681893, 5.05879144261738747321381003459, 6.06701711214470181316273645725, 6.58734948654215758674797991180, 7.24951787682004774617022100795, 8.262765729472943960745617735188, 8.571746181339135024403365704993

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