Properties

Label 2-2352-84.83-c0-0-0
Degree $2$
Conductor $2352$
Sign $0.188 - 0.981i$
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  − 3-s + 9-s + 1.73i·13-s − 19-s − 25-s − 27-s + 31-s + 37-s − 1.73i·39-s + 1.73i·43-s + 57-s + 1.73i·67-s + 1.73i·73-s + 75-s + 1.73i·79-s + ⋯
L(s)  = 1  − 3-s + 9-s + 1.73i·13-s − 19-s − 25-s − 27-s + 31-s + 37-s − 1.73i·39-s + 1.73i·43-s + 57-s + 1.73i·67-s + 1.73i·73-s + 75-s + 1.73i·79-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $0.188 - 0.981i$
Analytic conductor: \(1.17380\)
Root analytic conductor: \(1.08342\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (2351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :0),\ 0.188 - 0.981i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6974940110\)
\(L(\frac12)\) \(\approx\) \(0.6974940110\)
\(L(1)\) 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 \( 1 + T \)
7 \( 1 \)
good5 \( 1 + T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 - 1.73iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T + T^{2} \)
37 \( 1 - T + T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 - 1.73iT - T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.73iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - 1.73iT - T^{2} \)
79 \( 1 - 1.73iT - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.561270677503805122355498127439, −8.580616281445769283078165628490, −7.71305952099040439551703764777, −6.74082661981307605698247453557, −6.36745349331958052283105433998, −5.48990342263638676624315600201, −4.38516185517913566818394619067, −4.11968271461581993850576373944, −2.46380859867583483318890949769, −1.37557283099724187685215143061, 0.56972120470717488003431296185, 2.04506623390400058431833018549, 3.31429906632349366715179955226, 4.32072814752031782033117745321, 5.14378935307829044279684937249, 5.92008102714748470895582333305, 6.44846584921669495121452682915, 7.54067486386295911316731889946, 8.032578093317050911805226551568, 9.049289727976366684166746688305

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