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Properties

Label 2-1568-4.3-c2-0-28
Degree $2$
Conductor $1568$
Sign $0.707 - 0.707i$
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  + 5i·3-s − 9·5-s − 16·9-s − 3i·11-s − 16·13-s − 45i·15-s + 7·17-s − 11i·19-s − 19i·23-s + 56·25-s − 35i·27-s − 32·29-s + 11i·31-s + 15·33-s − 37-s + ⋯
L(s)  = 1  + 1.66i·3-s − 1.80·5-s − 1.77·9-s − 0.272i·11-s − 1.23·13-s − 3i·15-s + 0.411·17-s − 0.578i·19-s − 0.826i·23-s + 2.24·25-s − 1.29i·27-s − 1.10·29-s + 0.354i·31-s + 0.454·33-s − 0.0270·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $0.707 - 0.707i$
Analytic conductor: \(42.7249\)
Root analytic conductor: \(6.53642\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1),\ 0.707 - 0.707i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6728497609\)
\(L(\frac12)\) \(\approx\) \(0.6728497609\)
\(L(2)\) 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 \( 1 - 5iT - 9T^{2} \)
5 \( 1 + 9T + 25T^{2} \)
11 \( 1 + 3iT - 121T^{2} \)
13 \( 1 + 16T + 169T^{2} \)
17 \( 1 - 7T + 289T^{2} \)
19 \( 1 + 11iT - 361T^{2} \)
23 \( 1 + 19iT - 529T^{2} \)
29 \( 1 + 32T + 841T^{2} \)
31 \( 1 - 11iT - 961T^{2} \)
37 \( 1 + T + 1.36e3T^{2} \)
41 \( 1 - 40T + 1.68e3T^{2} \)
43 \( 1 + 40iT - 1.84e3T^{2} \)
47 \( 1 - 85iT - 2.20e3T^{2} \)
53 \( 1 - 7T + 2.80e3T^{2} \)
59 \( 1 - 53iT - 3.48e3T^{2} \)
61 \( 1 + 79T + 3.72e3T^{2} \)
67 \( 1 - 11iT - 4.48e3T^{2} \)
71 \( 1 + 48iT - 5.04e3T^{2} \)
73 \( 1 + 143T + 5.32e3T^{2} \)
79 \( 1 + 35iT - 6.24e3T^{2} \)
83 \( 1 - 8iT - 6.88e3T^{2} \)
89 \( 1 - 97T + 7.92e3T^{2} \)
97 \( 1 - 88T + 9.40e3T^{2} \)
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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

−9.241469677393549660983173810617, −8.770879447275554489682412083692, −7.76507993730426716718865338769, −7.23070544006016712593315010439, −5.83094642829043535980258202896, −4.71275087149820965822857292171, −4.44660887192312401649971201084, −3.50719770100004282061531214357, −2.80858471337558816284755212766, −0.35591139938768181442901152755, 0.56026313298801034541009365446, 1.82051250533610879457080468340, 2.97770790151589403408354670265, 3.93467931047523397666025701646, 5.04224514233906133356805699394, 6.10069978439213187211998301415, 7.20020540283511436664335860566, 7.47860959110570895089628262215, 7.939236293930860062248801817273, 8.806515657471662291110289769834

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