Properties

Label 2-1568-224.213-c0-0-0
Degree $2$
Conductor $1568$
Sign $0.792 - 0.610i$
Analytic cond. $0.782533$
Root an. cond. $0.884609$
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  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.607 − 0.465i)11-s + (0.500 + 0.866i)16-s + (0.499 − 0.866i)18-s + (0.292 + 0.707i)22-s + (1.36 + 0.366i)23-s + (0.965 − 0.258i)25-s + (0.707 − 0.292i)29-s + (−0.965 + 0.258i)32-s + (0.707 + 0.707i)36-s + (1.83 − 0.241i)37-s + (−1.70 − 0.707i)43-s + (−0.758 + 0.0999i)44-s + ⋯
L(s)  = 1  + (−0.258 + 0.965i)2-s + (−0.866 − 0.499i)4-s + (0.707 − 0.707i)8-s + (−0.965 − 0.258i)9-s + (0.607 − 0.465i)11-s + (0.500 + 0.866i)16-s + (0.499 − 0.866i)18-s + (0.292 + 0.707i)22-s + (1.36 + 0.366i)23-s + (0.965 − 0.258i)25-s + (0.707 − 0.292i)29-s + (−0.965 + 0.258i)32-s + (0.707 + 0.707i)36-s + (1.83 − 0.241i)37-s + (−1.70 − 0.707i)43-s + (−0.758 + 0.0999i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.792 - 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.792 - 0.610i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.792 - 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8787519168\)
\(L(\frac12)\) \(\approx\) \(0.8787519168\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 \)
good3 \( 1 + (0.965 + 0.258i)T^{2} \)
5 \( 1 + (-0.965 + 0.258i)T^{2} \)
11 \( 1 + (-0.607 + 0.465i)T + (0.258 - 0.965i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.258 + 0.965i)T^{2} \)
23 \( 1 + (-1.36 - 0.366i)T + (0.866 + 0.5i)T^{2} \)
29 \( 1 + (-0.707 + 0.292i)T + (0.707 - 0.707i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (-1.83 + 0.241i)T + (0.965 - 0.258i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + (-0.5 + 0.866i)T^{2} \)
53 \( 1 + (-1.46 + 1.12i)T + (0.258 - 0.965i)T^{2} \)
59 \( 1 + (0.258 - 0.965i)T^{2} \)
61 \( 1 + (-0.258 - 0.965i)T^{2} \)
67 \( 1 + (0.241 - 1.83i)T + (-0.965 - 0.258i)T^{2} \)
71 \( 1 + iT^{2} \)
73 \( 1 + (0.866 - 0.5i)T^{2} \)
79 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 + (0.866 + 0.5i)T^{2} \)
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

−9.450162849554007874914032433440, −8.677982151504933316922583223363, −8.337463247101296276148043804114, −7.16767979491051197464052362678, −6.55312750962969078391788771924, −5.73821658504702490517578023493, −4.99207299173732803927176203791, −3.92480582293908730913424133634, −2.84058506932663814556917294984, −0.981520388703506868513534067892, 1.19447181610287746628002711232, 2.56460838447072322396931691092, 3.25333407088804613811006807467, 4.48441999582263814272318543102, 5.11471690860405032697159664577, 6.30555176064252245121146380611, 7.29248232355138324120716989111, 8.256209748669277988355691807397, 8.886270384856537727774343177217, 9.505747797423482622652477082974

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