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Properties

Label 2-1568-8.3-c2-0-69
Degree $2$
Conductor $1568$
Sign $-0.929 + 0.369i$
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  + 0.0974·3-s − 3.46i·5-s − 8.99·9-s + 2.92·11-s − 19.1i·13-s − 0.337i·15-s + 14.3·17-s + 8.09·19-s + 16.7i·23-s + 12.9·25-s − 1.75·27-s + 27.1i·29-s − 44.8i·31-s + 0.285·33-s − 39.5i·37-s + ⋯
L(s)  = 1  + 0.0324·3-s − 0.693i·5-s − 0.998·9-s + 0.266·11-s − 1.47i·13-s − 0.0225i·15-s + 0.846·17-s + 0.426·19-s + 0.728i·23-s + 0.519·25-s − 0.0649·27-s + 0.936i·29-s − 1.44i·31-s + 0.00864·33-s − 1.06i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.929 + 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.929 + 0.369i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (687, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ -0.929 + 0.369i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.9290159413\)
\(L(\frac12)\) \(\approx\) \(0.9290159413\)
\(L(2)\) 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 \( 1 - 0.0974T + 9T^{2} \)
5 \( 1 + 3.46iT - 25T^{2} \)
11 \( 1 - 2.92T + 121T^{2} \)
13 \( 1 + 19.1iT - 169T^{2} \)
17 \( 1 - 14.3T + 289T^{2} \)
19 \( 1 - 8.09T + 361T^{2} \)
23 \( 1 - 16.7iT - 529T^{2} \)
29 \( 1 - 27.1iT - 841T^{2} \)
31 \( 1 + 44.8iT - 961T^{2} \)
37 \( 1 + 39.5iT - 1.36e3T^{2} \)
41 \( 1 + 45.8T + 1.68e3T^{2} \)
43 \( 1 + 61.0T + 1.84e3T^{2} \)
47 \( 1 - 46.2iT - 2.20e3T^{2} \)
53 \( 1 + 9.69iT - 2.80e3T^{2} \)
59 \( 1 + 114.T + 3.48e3T^{2} \)
61 \( 1 + 7.48iT - 3.72e3T^{2} \)
67 \( 1 - 12.0T + 4.48e3T^{2} \)
71 \( 1 + 129. iT - 5.04e3T^{2} \)
73 \( 1 - 18.2T + 5.32e3T^{2} \)
79 \( 1 + 42.6iT - 6.24e3T^{2} \)
83 \( 1 + 109.T + 6.88e3T^{2} \)
89 \( 1 - 80.9T + 7.92e3T^{2} \)
97 \( 1 + 162.T + 9.40e3T^{2} \)
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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

−8.907915196724077371212401012664, −8.055710094730170474119832555722, −7.57581345171539353963160234562, −6.27939769036303697720518913678, −5.47736773584207357593278972468, −4.98299599492581319993706882088, −3.57707346582749010917851065965, −2.91094145888172440205602784322, −1.40762863475086187553740160953, −0.25565376025543390265333967361, 1.48187388315270740483450452565, 2.73406243190462800825010280919, 3.46789711238360547495387822496, 4.61652515932444826184396221959, 5.53441623309433905878426640704, 6.60899715694044073171547817092, 6.90302711581056948535587370684, 8.157681976610368297118571116479, 8.712729488201792318082937209332, 9.625767285416406778509354366239

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