Properties

Label 2-84e2-21.20-c1-0-19
Degree $2$
Conductor $7056$
Sign $-0.192 - 0.981i$
Analytic cond. $56.3424$
Root an. cond. $7.50616$
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  − 2.93·5-s + 4.82i·11-s − 2.93i·13-s + 7.07·17-s + 5.86i·19-s − 2i·23-s + 3.58·25-s + 0.828i·29-s − 5.86i·31-s − 5.41·37-s + 1.21·41-s − 4.48·43-s + 5.86·47-s + 7.07i·53-s − 14.1i·55-s + ⋯
L(s)  = 1  − 1.31·5-s + 1.45i·11-s − 0.812i·13-s + 1.71·17-s + 1.34i·19-s − 0.417i·23-s + 0.717·25-s + 0.153i·29-s − 1.05i·31-s − 0.890·37-s + 0.189·41-s − 0.683·43-s + 0.854·47-s + 0.971i·53-s − 1.90i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.192 - 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.192 - 0.981i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (881, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ -0.192 - 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.065664010\)
\(L(\frac12)\) \(\approx\) \(1.065664010\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.93T + 5T^{2} \)
11 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 + 2.93iT - 13T^{2} \)
17 \( 1 - 7.07T + 17T^{2} \)
19 \( 1 - 5.86iT - 19T^{2} \)
23 \( 1 + 2iT - 23T^{2} \)
29 \( 1 - 0.828iT - 29T^{2} \)
31 \( 1 + 5.86iT - 31T^{2} \)
37 \( 1 + 5.41T + 37T^{2} \)
41 \( 1 - 1.21T + 41T^{2} \)
43 \( 1 + 4.48T + 43T^{2} \)
47 \( 1 - 5.86T + 47T^{2} \)
53 \( 1 - 7.07iT - 53T^{2} \)
59 \( 1 - 5.86T + 59T^{2} \)
61 \( 1 + 1.21iT - 61T^{2} \)
67 \( 1 - 8.48T + 67T^{2} \)
71 \( 1 - 0.828iT - 71T^{2} \)
73 \( 1 + 7.07iT - 73T^{2} \)
79 \( 1 + 1.65T + 79T^{2} \)
83 \( 1 + 11.7T + 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 - 7.07iT - 97T^{2} \)
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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

−7.909472561937407414031055688498, −7.60807728000517134218827097326, −7.00733516184432333331616375637, −5.94710343463148412496160771474, −5.29932510429385708256030943914, −4.44726249379117533325626059024, −3.78443863753041354592687700290, −3.18939853358993259507778788679, −2.05120183344966994532669281005, −0.936199096923585792646386629966, 0.34343656218144123503072271666, 1.27567932158929819208969679113, 2.72064048450740367774580464262, 3.50135839114853317794194900291, 3.88560367415907970986955049961, 4.96651550661391013181825978669, 5.50086298623551835165726038433, 6.48904837323152790168524016364, 7.13167535500710769324446678200, 7.74755663856340993266317330010

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