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Properties

Label 2-56e2-448.123-c0-0-0
Degree $2$
Conductor $3136$
Sign $0.999 + 0.0368i$
Analytic cond. $1.56506$
Root an. cond. $1.25102$
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  + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.382 + 0.923i)8-s + (0.608 − 0.793i)9-s + (0.128 + 1.95i)11-s + (0.866 − 0.5i)16-s + (−0.866 − 0.499i)18-s + (1.92 − 0.382i)22-s + (−1.12 + 1.46i)23-s + (0.793 − 0.608i)25-s + (−1.08 + 1.63i)29-s + (−0.608 − 0.793i)32-s + (−0.382 + 0.923i)36-s + (−1.05 + 0.357i)37-s + (0.923 + 1.38i)43-s + (−0.630 − 1.85i)44-s + ⋯
L(s)  = 1  + (−0.130 − 0.991i)2-s + (−0.965 + 0.258i)4-s + (0.382 + 0.923i)8-s + (0.608 − 0.793i)9-s + (0.128 + 1.95i)11-s + (0.866 − 0.5i)16-s + (−0.866 − 0.499i)18-s + (1.92 − 0.382i)22-s + (−1.12 + 1.46i)23-s + (0.793 − 0.608i)25-s + (−1.08 + 1.63i)29-s + (−0.608 − 0.793i)32-s + (−0.382 + 0.923i)36-s + (−1.05 + 0.357i)37-s + (0.923 + 1.38i)43-s + (−0.630 − 1.85i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3136\)    =    \(2^{6} \cdot 7^{2}\)
Sign: $0.999 + 0.0368i$
Analytic conductor: \(1.56506\)
Root analytic conductor: \(1.25102\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3136} (2811, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3136,\ (\ :0),\ 0.999 + 0.0368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9925202753\)
\(L(\frac12)\) \(\approx\) \(0.9925202753\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.130 + 0.991i)T \)
7 \( 1 \)
good3 \( 1 + (-0.608 + 0.793i)T^{2} \)
5 \( 1 + (-0.793 + 0.608i)T^{2} \)
11 \( 1 + (-0.128 - 1.95i)T + (-0.991 + 0.130i)T^{2} \)
13 \( 1 + (0.923 - 0.382i)T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T^{2} \)
19 \( 1 + (-0.130 + 0.991i)T^{2} \)
23 \( 1 + (1.12 - 1.46i)T + (-0.258 - 0.965i)T^{2} \)
29 \( 1 + (1.08 - 1.63i)T + (-0.382 - 0.923i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (1.05 - 0.357i)T + (0.793 - 0.608i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T^{2} \)
53 \( 1 + (-1.65 + 0.108i)T + (0.991 - 0.130i)T^{2} \)
59 \( 1 + (0.130 + 0.991i)T^{2} \)
61 \( 1 + (-0.991 - 0.130i)T^{2} \)
67 \( 1 + (0.491 + 0.996i)T + (-0.608 + 0.793i)T^{2} \)
71 \( 1 + (-1.30 - 0.541i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.965 + 0.258i)T^{2} \)
79 \( 1 + (-0.739 - 0.198i)T + (0.866 + 0.5i)T^{2} \)
83 \( 1 + (0.923 - 0.382i)T^{2} \)
89 \( 1 + (-0.965 + 0.258i)T^{2} \)
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

−9.321783271727610934129175026974, −8.238297830841118121593022584731, −7.35024359142200721084610523446, −6.86360539871523165315352460278, −5.60728893335158161584465412307, −4.74829896605771702639579767710, −4.06481426773489766036368177617, −3.32422934557082885109676135232, −2.08512015347979654403087006130, −1.38463706298877423420351589373, 0.69520030522398274853399270684, 2.24186605205285727157963461098, 3.62958625525482982179803932141, 4.24720330871027903365773801891, 5.33418148434663864880758733018, 5.83215327282920421232494813587, 6.61543085171591856766029012356, 7.42520391402463755470312132752, 8.137915727997676779583371965549, 8.660962583107742119034215089784

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