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Properties

Label 2-1568-7.6-c2-0-51
Degree $2$
Conductor $1568$
Sign $0.755 + 0.654i$
Analytic cond. $42.7249$
Root an. cond. $6.53642$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.506i·3-s − 5.30i·5-s + 8.74·9-s + 17.0·11-s − 21.4i·13-s + 2.68·15-s + 24.0i·17-s + 12.2i·19-s + 40.2·23-s − 3.11·25-s + 8.99i·27-s − 26.0·29-s − 25.2i·31-s + 8.66i·33-s + 12.9·37-s + ⋯
L(s)  = 1  + 0.168i·3-s − 1.06i·5-s + 0.971·9-s + 1.55·11-s − 1.65i·13-s + 0.179·15-s + 1.41i·17-s + 0.643i·19-s + 1.75·23-s − 0.124·25-s + 0.333i·27-s − 0.899·29-s − 0.814i·31-s + 0.262i·33-s + 0.350·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.573798072\)
\(L(\frac12)\) \(\approx\) \(2.573798072\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 \)
good3 \( 1 - 0.506iT - 9T^{2} \)
5 \( 1 + 5.30iT - 25T^{2} \)
11 \( 1 - 17.0T + 121T^{2} \)
13 \( 1 + 21.4iT - 169T^{2} \)
17 \( 1 - 24.0iT - 289T^{2} \)
19 \( 1 - 12.2iT - 361T^{2} \)
23 \( 1 - 40.2T + 529T^{2} \)
29 \( 1 + 26.0T + 841T^{2} \)
31 \( 1 + 25.2iT - 961T^{2} \)
37 \( 1 - 12.9T + 1.36e3T^{2} \)
41 \( 1 - 33.8iT - 1.68e3T^{2} \)
43 \( 1 + 29.9T + 1.84e3T^{2} \)
47 \( 1 - 55.7iT - 2.20e3T^{2} \)
53 \( 1 + 8.72T + 2.80e3T^{2} \)
59 \( 1 + 49.9iT - 3.48e3T^{2} \)
61 \( 1 - 3.93iT - 3.72e3T^{2} \)
67 \( 1 - 105.T + 4.48e3T^{2} \)
71 \( 1 - 35.0T + 5.04e3T^{2} \)
73 \( 1 + 46.6iT - 5.32e3T^{2} \)
79 \( 1 + 86.9T + 6.24e3T^{2} \)
83 \( 1 + 64.0iT - 6.88e3T^{2} \)
89 \( 1 - 42.9iT - 7.92e3T^{2} \)
97 \( 1 + 28.7iT - 9.40e3T^{2} \)
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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

−9.219303745669668562947603962045, −8.381313384175493699047433338393, −7.72820172918423222862099785423, −6.67135375062244985550593382348, −5.83441197208157192560669757092, −4.93552649496519718863669784706, −4.10894564803466221634957787451, −3.33662156794051853534453554649, −1.58996917410598805725031735199, −0.914953137891084150327055173891, 1.10018686681295762715658794370, 2.19525039179471744862896216783, 3.34865683706350882320007957308, 4.21991188928832077422624108505, 5.09603413235126942075674339150, 6.57618989429645006910541016796, 6.98432187775268850569612657693, 7.18119569780022166520356991540, 8.845522504617096572257486833627, 9.270811268229954371839897873320

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