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Properties

Label 2-1568-32.27-c0-0-0
Degree $2$
Conductor $1568$
Sign $0.980 - 0.195i$
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  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 + 0.292i)11-s − 1.00·16-s + 1.00·18-s + (−0.292 − 0.707i)22-s + (1 + i)23-s + (−0.707 − 0.707i)25-s + (0.707 + 1.70i)29-s + (0.707 + 0.707i)32-s + (−0.707 − 0.707i)36-s + (−0.707 − 0.292i)37-s + (1.70 + 0.707i)43-s + (−0.292 + 0.707i)44-s + ⋯
L(s)  = 1  + (−0.707 − 0.707i)2-s + 1.00i·4-s + (0.707 − 0.707i)8-s + (−0.707 + 0.707i)9-s + (0.707 + 0.292i)11-s − 1.00·16-s + 1.00·18-s + (−0.292 − 0.707i)22-s + (1 + i)23-s + (−0.707 − 0.707i)25-s + (0.707 + 1.70i)29-s + (0.707 + 0.707i)32-s + (−0.707 − 0.707i)36-s + (−0.707 − 0.292i)37-s + (1.70 + 0.707i)43-s + (−0.292 + 0.707i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.980 - 0.195i$
Analytic conductor: \(0.782533\)
Root analytic conductor: \(0.884609\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1275, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :0),\ 0.980 - 0.195i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7368074674\)
\(L(\frac12)\) \(\approx\) \(0.7368074674\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 0.707i)T \)
7 \( 1 \)
good3 \( 1 + (0.707 - 0.707i)T^{2} \)
5 \( 1 + (0.707 + 0.707i)T^{2} \)
11 \( 1 + (-0.707 - 0.292i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.707 - 0.707i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + (-1 - i)T + iT^{2} \)
29 \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \)
41 \( 1 - iT^{2} \)
43 \( 1 + (-1.70 - 0.707i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (-0.707 - 0.707i)T^{2} \)
61 \( 1 + (-0.707 + 0.707i)T^{2} \)
67 \( 1 + (-1.70 + 0.707i)T + (0.707 - 0.707i)T^{2} \)
71 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
73 \( 1 - iT^{2} \)
79 \( 1 - 1.41T + T^{2} \)
83 \( 1 + (-0.707 + 0.707i)T^{2} \)
89 \( 1 + iT^{2} \)
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

−9.529034014802124882671447288175, −9.000032768279324597756800357139, −8.224696830817156949627853157165, −7.44391272844857297793611666791, −6.66640746776151022767153792532, −5.46892101243739222507119211348, −4.48005800797459833174797614894, −3.43879139777043829952931053248, −2.51712324361124875632430646967, −1.36601141306477706755349510952, 0.830273354711556467065703469392, 2.36621571555812480692928309701, 3.68892987091514333311053187165, 4.79903656229115455758747502400, 5.84485244555263839439400979653, 6.37757125908925135101429686566, 7.18011948530548228231016023838, 8.127188706170689416193556396965, 8.821623064266671119529435900069, 9.357794394836919016888598388498

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