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Properties

Label 2-84e2-21.20-c1-0-28
Degree $2$
Conductor $7056$
Sign $-0.0980 - 0.995i$
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  + 2.44·5-s + 1.41i·11-s + 5.19i·13-s + 4.89·17-s + 1.73i·19-s + 5.65i·23-s + 0.999·25-s + 2.82i·29-s − 1.73i·31-s − 37-s − 7.34·41-s + 43-s + 12.2·47-s + 2.82i·53-s + 3.46i·55-s + ⋯
L(s)  = 1  + 1.09·5-s + 0.426i·11-s + 1.44i·13-s + 1.18·17-s + 0.397i·19-s + 1.17i·23-s + 0.199·25-s + 0.525i·29-s − 0.311i·31-s − 0.164·37-s − 1.14·41-s + 0.152·43-s + 1.78·47-s + 0.388i·53-s + 0.467i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-0.0980 - 0.995i$
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: \((2,\ 7056,\ (\ :1/2),\ -0.0980 - 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.230330529\)
\(L(\frac12)\) \(\approx\) \(2.230330529\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.44T + 5T^{2} \)
11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 - 4.89T + 17T^{2} \)
19 \( 1 - 1.73iT - 19T^{2} \)
23 \( 1 - 5.65iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + T + 37T^{2} \)
41 \( 1 + 7.34T + 41T^{2} \)
43 \( 1 - T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 - 2.82iT - 53T^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 - 3.46iT - 61T^{2} \)
67 \( 1 + 11T + 67T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 - 1.73iT - 73T^{2} \)
79 \( 1 + 5T + 79T^{2} \)
83 \( 1 + 7.34T + 83T^{2} \)
89 \( 1 + 4.89T + 89T^{2} \)
97 \( 1 - 10.3iT - 97T^{2} \)
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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

−8.074282545263447465982225630250, −7.31284489148166668714586338125, −6.78284442960534553010042640625, −5.83516588679637268812332139751, −5.55081411342743932665431190652, −4.57642842218683057459276027190, −3.81233192804657634175012320507, −2.87841438949152355018579044924, −1.85005302828345472760920941886, −1.39017117966199015289512597999, 0.51677079584257980954600342333, 1.50672230739022740465729408739, 2.61118238528323154424405680110, 3.12169357854006257879206873459, 4.15611499548976281431704925393, 5.18942067637388472590818809646, 5.63771099648837666345027695026, 6.18808629711813713840962457157, 7.02444245549940855551163194234, 7.83158979239588019437287935672

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