Properties

Label 2-1568-7.4-c1-0-33
Degree $2$
Conductor $1568$
Sign $-0.386 + 0.922i$
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 − 1.73i)3-s + (−0.499 − 0.866i)9-s + (2 − 3.46i)11-s − 4·13-s + (1 − 1.73i)17-s + (3 + 5.19i)19-s + (−4 − 6.92i)23-s + (2.5 − 4.33i)25-s + 4.00·27-s + 2·29-s + (2 − 3.46i)31-s + (−3.99 − 6.92i)33-s + (−5 − 8.66i)37-s + (−4 + 6.92i)39-s − 10·41-s + ⋯
L(s)  = 1  + (0.577 − 0.999i)3-s + (−0.166 − 0.288i)9-s + (0.603 − 1.04i)11-s − 1.10·13-s + (0.242 − 0.420i)17-s + (0.688 + 1.19i)19-s + (−0.834 − 1.44i)23-s + (0.5 − 0.866i)25-s + 0.769·27-s + 0.371·29-s + (0.359 − 0.622i)31-s + (−0.696 − 1.20i)33-s + (−0.821 − 1.42i)37-s + (−0.640 + 1.10i)39-s − 1.56·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.386 + 0.922i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.902213837\)
\(L(\frac12)\) \(\approx\) \(1.902213837\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 1.73i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-1 + 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 5.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4 + 6.92i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (5 + 8.66i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (2 + 3.46i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1 + 1.73i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 - 6.92i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-4 + 6.92i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (-3 + 5.19i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8 - 13.8i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 2T + 83T^{2} \)
89 \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−8.935618744639606178158548065259, −8.288160011650115988026063046662, −7.66673133820314280638966525338, −6.84796716705882411980144600197, −6.13014693493831836706981825908, −5.10182222428006508201329252193, −3.95419917458395826946989809104, −2.86626175830223330806690385101, −2.01399891433192791332222017452, −0.69972999514909345743633137145, 1.59508245037601386228854357373, 2.93991732126155711150269897491, 3.69065127170005976181181153096, 4.74148336838648099636782402493, 5.16370727224857433240921688883, 6.65495064580876221518634699169, 7.24502425935227376474024920526, 8.226742755290609853996172103233, 9.131990479126547086206410105317, 9.764664668741302425296577013520

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