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Properties

Label 2-84e2-28.27-c1-0-59
Degree $2$
Conductor $7056$
Sign $0.944 - 0.327i$
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  + 3.46i·5-s − 3.46i·11-s + 5.19i·13-s − 6.92i·17-s + 7·19-s − 6.99·25-s + 5·31-s + 37-s − 10.3i·41-s + 1.73i·43-s + 6·47-s + 11.9·55-s − 18·65-s − 1.73i·67-s + 3.46i·71-s + ⋯
L(s)  = 1  + 1.54i·5-s − 1.04i·11-s + 1.44i·13-s − 1.68i·17-s + 1.60·19-s − 1.39·25-s + 0.898·31-s + 0.164·37-s − 1.62i·41-s + 0.264i·43-s + 0.875·47-s + 1.61·55-s − 2.23·65-s − 0.211i·67-s + 0.411i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(7056\)    =    \(2^{4} \cdot 3^{2} \cdot 7^{2}\)
Sign: $0.944 - 0.327i$
Analytic conductor: \(56.3424\)
Root analytic conductor: \(7.50616\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{7056} (1567, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 7056,\ (\ :1/2),\ 0.944 - 0.327i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.087530992\)
\(L(\frac12)\) \(\approx\) \(2.087530992\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.46iT - 5T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 - 5.19iT - 13T^{2} \)
17 \( 1 + 6.92iT - 17T^{2} \)
19 \( 1 - 7T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 - T + 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 1.73iT - 43T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 1.73iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + 8.66iT - 73T^{2} \)
79 \( 1 + 15.5iT - 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−7.65611881761990967708807069693, −7.26697735051634026350701299758, −6.67815663342761930656247052334, −6.01463174764086440807617711324, −5.23699617192387628994069607111, −4.31562701063703723150733037427, −3.33429489693943252873393223599, −2.94111092834589446620875486245, −2.05546043617351061551266639194, −0.67987550782813973866333310147, 0.871342897843147157232852418468, 1.47399982517872016891569393041, 2.64004020004063021094859250249, 3.66760424636713438528294124816, 4.41389453316476277063711347858, 5.13075357967502039251074227677, 5.58821386465766116908128869490, 6.39155363510499330159414877566, 7.44398135024813468885859670508, 8.040187477939513771311360637997

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