Properties

Degree $2$
Conductor $7056$
Sign $-0.944 + 0.327i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 3.87i·5-s − 3.87i·11-s − 3.46i·13-s + 4·19-s − 7.74i·23-s − 10.0·25-s + 6.70·29-s − 31-s + 4·37-s − 7.74i·41-s − 6.92i·43-s + 13.4·47-s − 6.70·53-s − 15.0·55-s + 6.70·59-s + ⋯
L(s)  = 1  − 1.73i·5-s − 1.16i·11-s − 0.960i·13-s + 0.917·19-s − 1.61i·23-s − 2.00·25-s + 1.24·29-s − 0.179·31-s + 0.657·37-s − 1.20i·41-s − 1.05i·43-s + 1.95·47-s − 0.921·53-s − 2.02·55-s + 0.873·59-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: $-0.944 + 0.327i$
Motivic weight: \(1\)
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

\(L(1)\) \(\approx\) \(2.015908699\)
\(L(\frac12)\) \(\approx\) \(2.015908699\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.87iT - 5T^{2} \)
11 \( 1 + 3.87iT - 11T^{2} \)
13 \( 1 + 3.46iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + 7.74iT - 23T^{2} \)
29 \( 1 - 6.70T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + 7.74iT - 41T^{2} \)
43 \( 1 + 6.92iT - 43T^{2} \)
47 \( 1 - 13.4T + 47T^{2} \)
53 \( 1 + 6.70T + 53T^{2} \)
59 \( 1 - 6.70T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 + 6.92iT - 67T^{2} \)
71 \( 1 - 7.74iT - 71T^{2} \)
73 \( 1 - 6.92iT - 73T^{2} \)
79 \( 1 + 12.1iT - 79T^{2} \)
83 \( 1 - 6.70T + 83T^{2} \)
89 \( 1 + 7.74iT - 89T^{2} \)
97 \( 1 - 5.19iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.86776925416005717170718549626, −6.99526255573787874388771538616, −5.87173599178948858889300627583, −5.60357944718570714092716278009, −4.80644378787604916477876619009, −4.15482889882680533426857102683, −3.23744140325021410495133196844, −2.30549641026596195091709438638, −0.908493481907067321896044956712, −0.63329011669784976241150922127, 1.42725942948287277683751612130, 2.35646533275950541110443070020, 3.03865359479527171503023370452, 3.82699095089392743476653733585, 4.61173007060716447557845031909, 5.51979078214773733092702489143, 6.46461448201961730276870472633, 6.76133887599449272204996014560, 7.58973606945621124778237268265, 7.82189768560627078753388875684

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