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Properties

Label 2-1568-8.3-c2-0-30
Degree $2$
Conductor $1568$
Sign $0.527 - 0.849i$
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  − 3.44·3-s + 4.88i·5-s + 2.84·9-s + 21.4·11-s + 13.0i·13-s − 16.8i·15-s + 0.234·17-s + 4.55·19-s − 10.9i·23-s + 1.15·25-s + 21.1·27-s + 34.6i·29-s − 34.1i·31-s − 73.9·33-s − 54.2i·37-s + ⋯
L(s)  = 1  − 1.14·3-s + 0.976i·5-s + 0.315·9-s + 1.95·11-s + 1.00i·13-s − 1.12i·15-s + 0.0138·17-s + 0.239·19-s − 0.476i·23-s + 0.0463·25-s + 0.784·27-s + 1.19i·29-s − 1.10i·31-s − 2.23·33-s − 1.46i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.527 - 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.527 - 0.849i$
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: \((2,\ 1568,\ (\ :1),\ 0.527 - 0.849i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.439185623\)
\(L(\frac12)\) \(\approx\) \(1.439185623\)
\(L(2)\) 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 \( 1 + 3.44T + 9T^{2} \)
5 \( 1 - 4.88iT - 25T^{2} \)
11 \( 1 - 21.4T + 121T^{2} \)
13 \( 1 - 13.0iT - 169T^{2} \)
17 \( 1 - 0.234T + 289T^{2} \)
19 \( 1 - 4.55T + 361T^{2} \)
23 \( 1 + 10.9iT - 529T^{2} \)
29 \( 1 - 34.6iT - 841T^{2} \)
31 \( 1 + 34.1iT - 961T^{2} \)
37 \( 1 + 54.2iT - 1.36e3T^{2} \)
41 \( 1 - 37.8T + 1.68e3T^{2} \)
43 \( 1 - 4.84T + 1.84e3T^{2} \)
47 \( 1 + 72.3iT - 2.20e3T^{2} \)
53 \( 1 - 21.6iT - 2.80e3T^{2} \)
59 \( 1 - 34.9T + 3.48e3T^{2} \)
61 \( 1 + 63.6iT - 3.72e3T^{2} \)
67 \( 1 + 18.4T + 4.48e3T^{2} \)
71 \( 1 - 47.5iT - 5.04e3T^{2} \)
73 \( 1 + 55.9T + 5.32e3T^{2} \)
79 \( 1 - 95.0iT - 6.24e3T^{2} \)
83 \( 1 - 71.5T + 6.88e3T^{2} \)
89 \( 1 - 159.T + 7.92e3T^{2} \)
97 \( 1 - 90.4T + 9.40e3T^{2} \)
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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

−9.328698397338527119397529704065, −8.839403162982973839295310295065, −7.42219130610072039767225762891, −6.64675078015890057076443295014, −6.39162702984241644718508422633, −5.41466718006719401611381155574, −4.30876252701644644928065446376, −3.55923487833135072458006534675, −2.17072398210580240870111777372, −0.870476529755248046483558480635, 0.69746549871816741035534881510, 1.37734873626061909885696098580, 3.17444230441956847707587929551, 4.33413975602141311261000292603, 4.98442545388875228654414347420, 5.92079394322073711720688986466, 6.38686001696288721152008896280, 7.44148455573576330087826211082, 8.463918547217296465862210771783, 9.108435706284560373953514034744

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