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Properties

Label 2-84e2-21.20-c1-0-39
Degree $2$
Conductor $7056$
Sign $0.970 - 0.239i$
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  + 0.717·5-s − 3i·11-s + 2.44i·13-s + 5.91·17-s + 5.91i·19-s − 4.24i·23-s − 4.48·25-s + 7.24i·29-s − 9.08i·31-s − 0.242·37-s + 11.8·41-s + 0.242·43-s − 5.91·47-s + 7.24i·53-s − 2.15i·55-s + ⋯
L(s)  = 1  + 0.320·5-s − 0.904i·11-s + 0.679i·13-s + 1.43·17-s + 1.35i·19-s − 0.884i·23-s − 0.897·25-s + 1.34i·29-s − 1.63i·31-s − 0.0398·37-s + 1.84·41-s + 0.0370·43-s − 0.862·47-s + 0.994i·53-s − 0.290i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 - 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.970 - 0.239i$
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: \((2,\ 7056,\ (\ :1/2),\ 0.970 - 0.239i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.200046906\)
\(L(\frac12)\) \(\approx\) \(2.200046906\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.717T + 5T^{2} \)
11 \( 1 + 3iT - 11T^{2} \)
13 \( 1 - 2.44iT - 13T^{2} \)
17 \( 1 - 5.91T + 17T^{2} \)
19 \( 1 - 5.91iT - 19T^{2} \)
23 \( 1 + 4.24iT - 23T^{2} \)
29 \( 1 - 7.24iT - 29T^{2} \)
31 \( 1 + 9.08iT - 31T^{2} \)
37 \( 1 + 0.242T + 37T^{2} \)
41 \( 1 - 11.8T + 41T^{2} \)
43 \( 1 - 0.242T + 43T^{2} \)
47 \( 1 + 5.91T + 47T^{2} \)
53 \( 1 - 7.24iT - 53T^{2} \)
59 \( 1 - 8.06T + 59T^{2} \)
61 \( 1 - 1.01iT - 61T^{2} \)
67 \( 1 - 10T + 67T^{2} \)
71 \( 1 - 1.75iT - 71T^{2} \)
73 \( 1 - 1.43iT - 73T^{2} \)
79 \( 1 - 2.75T + 79T^{2} \)
83 \( 1 + 6.63T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 + 13.5iT - 97T^{2} \)
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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

−7.949549631543060558579561880902, −7.40178906351288472849955582304, −6.39213447868494801306739878555, −5.86102030555956092924136923804, −5.36582485953828354355700684498, −4.23053965716593619645422736435, −3.66456337620449242037908364184, −2.76737952802638415738256398601, −1.80035373753227269061797918010, −0.838318922216075097983801542521, 0.70958254669632545335344746530, 1.78488839669787893384490655466, 2.68572115816442472661767135595, 3.47950189876686875484464079628, 4.36771686339384526760028714452, 5.22263146620710600101849003613, 5.65885233081639516133178025845, 6.56836035194746369195629633849, 7.29021731622609340871497528101, 7.83016014299990476597126054553

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