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Properties

Label 2-1568-7.2-c1-0-4
Degree $2$
Conductor $1568$
Sign $-0.968 + 0.250i$
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  + (1.61 + 2.80i)3-s + (−0.618 + 1.07i)5-s + (−3.73 + 6.47i)9-s + (1.23 + 2.14i)11-s − 5.23·13-s − 4·15-s + (−2.23 − 3.87i)17-s + (−1.61 + 2.80i)19-s + (2 − 3.46i)23-s + (1.73 + 3.00i)25-s − 14.4·27-s + 4.47·29-s + (−3.23 − 5.60i)31-s + (−3.99 + 6.92i)33-s + (−2.23 + 3.87i)37-s + ⋯
L(s)  = 1  + (0.934 + 1.61i)3-s + (−0.276 + 0.478i)5-s + (−1.24 + 2.15i)9-s + (0.372 + 0.645i)11-s − 1.45·13-s − 1.03·15-s + (−0.542 − 0.939i)17-s + (−0.371 + 0.642i)19-s + (0.417 − 0.722i)23-s + (0.347 + 0.601i)25-s − 2.78·27-s + 0.830·29-s + (−0.581 − 1.00i)31-s + (−0.696 + 1.20i)33-s + (−0.367 + 0.636i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.968 + 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.968 + 0.250i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.489393601\)
\(L(\frac12)\) \(\approx\) \(1.489393601\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.61 - 2.80i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (0.618 - 1.07i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.23 - 2.14i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.23T + 13T^{2} \)
17 \( 1 + (2.23 + 3.87i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.61 - 2.80i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + (3.23 + 5.60i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.23 - 3.87i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 0.472T + 41T^{2} \)
43 \( 1 - 2.47T + 43T^{2} \)
47 \( 1 + (0.763 - 1.32i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5 - 8.66i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.38 - 4.12i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.38 + 5.85i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12.9T + 71T^{2} \)
73 \( 1 + (-7.47 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.47 - 4.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 4.76T + 83T^{2} \)
89 \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 3.52T + 97T^{2} \)
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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

−9.808867553959873369870795087391, −9.232435568790009891046173794398, −8.489706144804307061619366564210, −7.55703358645982507693548620579, −6.86859692137323886754057589843, −5.40982237375560204715947406723, −4.61252942471328041322833687589, −4.07298506239955432302135653915, −2.94483496716764376380930812523, −2.35176383257612582945546986161, 0.48359294668717288006358622253, 1.72169460074768065049213250980, 2.63443828769953220830689879128, 3.59624293226686595412608707236, 4.82292299961505051289039086155, 5.99571283095785953892577339579, 6.88489430914071804396059957672, 7.31986299220303458610311275235, 8.345680336297989133915197461485, 8.665505201660069572403714629549

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