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Properties

Label 2-1568-56.3-c1-0-33
Degree $2$
Conductor $1568$
Sign $-0.982 + 0.185i$
Analytic cond. $12.5205$
Root an. cond. $3.53843$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.662 + 0.382i)3-s + (−1.51 − 2.62i)5-s + (−1.20 − 2.09i)9-s + (2.12 − 3.67i)11-s − 3.02·13-s − 2.31i·15-s + (3.86 + 2.23i)17-s + (−2.92 + 1.68i)19-s + (−4.84 + 2.79i)23-s + (−2.08 + 3.61i)25-s − 4.14i·27-s − 5.59i·29-s + (−5.16 + 8.95i)31-s + (2.81 − 1.62i)33-s + (−2.00 + 1.15i)37-s + ⋯
L(s)  = 1  + (0.382 + 0.220i)3-s + (−0.677 − 1.17i)5-s + (−0.402 − 0.696i)9-s + (0.639 − 1.10i)11-s − 0.839·13-s − 0.598i·15-s + (0.936 + 0.540i)17-s + (−0.671 + 0.387i)19-s + (−1.01 + 0.583i)23-s + (−0.417 + 0.722i)25-s − 0.797i·27-s − 1.03i·29-s + (−0.928 + 1.60i)31-s + (0.489 − 0.282i)33-s + (−0.330 + 0.190i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 + 0.185i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.982 + 0.185i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -0.982 + 0.185i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6898285093\)
\(L(\frac12)\) \(\approx\) \(0.6898285093\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.662 - 0.382i)T + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.51 + 2.62i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2.12 + 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 3.02T + 13T^{2} \)
17 \( 1 + (-3.86 - 2.23i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.92 - 1.68i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.84 - 2.79i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 5.59iT - 29T^{2} \)
31 \( 1 + (5.16 - 8.95i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.00 - 1.15i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 7.07iT - 41T^{2} \)
43 \( 1 + 2.58T + 43T^{2} \)
47 \( 1 + (3.02 + 5.24i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.83 - 1.63i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.32 + 5.38i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.65 + 6.33i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1 - 1.73i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 14.4iT - 71T^{2} \)
73 \( 1 + (2.92 + 1.68i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-9.69 + 5.59i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 9.23iT - 83T^{2} \)
89 \( 1 + (-2.14 + 1.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 6.88iT - 97T^{2} \)
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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

−9.005171485162979232662911759346, −8.241675519359137205519788741913, −7.88474327858437483431466908116, −6.51381697738005310327287665894, −5.75273385331644759154605087256, −4.78743376479830489761867962118, −3.81810159539815435198992132862, −3.26987840346154809937495389414, −1.58050249651185448884700880674, −0.24663599305622618382498777735, 2.00354681174829580671823621106, 2.75786744897993826134391021544, 3.79084549496865630111792830297, 4.70378760076296068456324005406, 5.79652186668010038329891118982, 6.96554810682672312376899078079, 7.33810208528034281358547523066, 7.974283713019662042298808488551, 9.007922473651500994807028917398, 9.854269340704915971596098786168

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