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Properties

Label 2-2352-4.3-c2-0-46
Degree $2$
Conductor $2352$
Sign $-i$
Analytic cond. $64.0873$
Root an. cond. $8.00545$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.73i·3-s + 7.41·5-s − 2.99·9-s + 20.1i·11-s + 21.8·13-s + 12.8i·15-s + 13.0·17-s + 14.6i·19-s + 40.9i·23-s + 29.9·25-s − 5.19i·27-s + 32.4·29-s − 51.0i·31-s − 34.9·33-s − 33.9·37-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.48·5-s − 0.333·9-s + 1.83i·11-s + 1.68·13-s + 0.856i·15-s + 0.768·17-s + 0.773i·19-s + 1.78i·23-s + 1.19·25-s − 0.192i·27-s + 1.12·29-s − 1.64i·31-s − 1.05·33-s − 0.917·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-i$
Analytic conductor: \(64.0873\)
Root analytic conductor: \(8.00545\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1),\ -i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(3.184029618\)
\(L(\frac12)\) \(\approx\) \(3.184029618\)
\(L(2)\) 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 \( 1 - 1.73iT \)
7 \( 1 \)
good5 \( 1 - 7.41T + 25T^{2} \)
11 \( 1 - 20.1iT - 121T^{2} \)
13 \( 1 - 21.8T + 169T^{2} \)
17 \( 1 - 13.0T + 289T^{2} \)
19 \( 1 - 14.6iT - 361T^{2} \)
23 \( 1 - 40.9iT - 529T^{2} \)
29 \( 1 - 32.4T + 841T^{2} \)
31 \( 1 + 51.0iT - 961T^{2} \)
37 \( 1 + 33.9T + 1.36e3T^{2} \)
41 \( 1 + 8.10T + 1.68e3T^{2} \)
43 \( 1 + 69.7iT - 1.84e3T^{2} \)
47 \( 1 - 28.5iT - 2.20e3T^{2} \)
53 \( 1 + 72.9T + 2.80e3T^{2} \)
59 \( 1 - 4.40iT - 3.48e3T^{2} \)
61 \( 1 + 33.9T + 3.72e3T^{2} \)
67 \( 1 + 30.5iT - 4.48e3T^{2} \)
71 \( 1 + 27.6iT - 5.04e3T^{2} \)
73 \( 1 - 61.4T + 5.32e3T^{2} \)
79 \( 1 - 90.5iT - 6.24e3T^{2} \)
83 \( 1 - 5.24iT - 6.88e3T^{2} \)
89 \( 1 + 56.8T + 7.92e3T^{2} \)
97 \( 1 - 123.T + 9.40e3T^{2} \)
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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

−9.356922160522382906996607035002, −8.326979823216361793145194245621, −7.46820820457619826148055526279, −6.46926196822936488501667879314, −5.78635057915448746967028664781, −5.20782543475960376107016141642, −4.16592670963345229995888599773, −3.31501368494993253053685164023, −2.01837130752060678779952259649, −1.40925421513066157818990240557, 0.805721318105038327472245186024, 1.48243426222085940900080785820, 2.78486865042626087058296151810, 3.35949964832535468312225887976, 4.86893788847150908126870829394, 5.70799624459140453850516729126, 6.38425469648322560692398133260, 6.60333378963298373715989401654, 8.134131449681079486836271990718, 8.621896893822937793830664034278

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