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Properties

Label 2-2352-21.20-c1-0-51
Degree $2$
Conductor $2352$
Sign $0.370 + 0.928i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.53 − 0.795i)3-s + 3.80·5-s + (1.73 + 2.44i)9-s − 0.357i·11-s − 4.04i·13-s + (−5.84 − 3.02i)15-s + 0.103·17-s + 2.45i·19-s − 1.33i·23-s + 9.44·25-s + (−0.723 − 5.14i)27-s − 4.97i·29-s − 7.88i·31-s + (−0.284 + 0.549i)33-s − 10.9·37-s + ⋯
L(s)  = 1  + (−0.888 − 0.459i)3-s + 1.69·5-s + (0.578 + 0.815i)9-s − 0.107i·11-s − 1.12i·13-s + (−1.50 − 0.780i)15-s + 0.0252·17-s + 0.563i·19-s − 0.277i·23-s + 1.88·25-s + (−0.139 − 0.990i)27-s − 0.923i·29-s − 1.41i·31-s + (−0.0494 + 0.0957i)33-s − 1.79·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.370 + 0.928i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ 0.370 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.784025778\)
\(L(\frac12)\) \(\approx\) \(1.784025778\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.53 + 0.795i)T \)
7 \( 1 \)
good5 \( 1 - 3.80T + 5T^{2} \)
11 \( 1 + 0.357iT - 11T^{2} \)
13 \( 1 + 4.04iT - 13T^{2} \)
17 \( 1 - 0.103T + 17T^{2} \)
19 \( 1 - 2.45iT - 19T^{2} \)
23 \( 1 + 1.33iT - 23T^{2} \)
29 \( 1 + 4.97iT - 29T^{2} \)
31 \( 1 + 7.88iT - 31T^{2} \)
37 \( 1 + 10.9T + 37T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 + 0.502T + 43T^{2} \)
47 \( 1 - 11.4T + 47T^{2} \)
53 \( 1 + 5.86iT - 53T^{2} \)
59 \( 1 - 7.54T + 59T^{2} \)
61 \( 1 - 9.47iT - 61T^{2} \)
67 \( 1 + 2.68T + 67T^{2} \)
71 \( 1 - 5.78iT - 71T^{2} \)
73 \( 1 - 0.235iT - 73T^{2} \)
79 \( 1 + 3.22T + 79T^{2} \)
83 \( 1 - 9.07T + 83T^{2} \)
89 \( 1 + 6.82T + 89T^{2} \)
97 \( 1 - 5.14iT - 97T^{2} \)
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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

−8.902396667767304028928323900702, −7.982057784339710923936641093141, −7.14529781538152643218744087826, −6.27767793608148530916298562143, −5.67137916204050874406231829232, −5.35244742716204912801934888164, −4.13748531653234237809048912114, −2.66038412285582785333778277143, −1.88115341868601045010403604718, −0.73609134197326094903676878877, 1.24717333038836533399089220726, 2.15953960976355953842201159308, 3.44039350287403411356879735345, 4.63567051089703567964398341326, 5.23679616994836697046294586199, 5.93675886603421739369862356490, 6.70132631466172629628246463479, 7.17188362153064838435443351069, 8.917332199927033556287688269484, 9.087777016090649558877852377398

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