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Properties

Label 2-1568-7.4-c1-0-29
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  + (−0.618 + 1.07i)3-s + (−1.61 − 2.80i)5-s + (0.736 + 1.27i)9-s + (3.23 − 5.60i)11-s + 0.763·13-s + 4·15-s + (−2.23 + 3.87i)17-s + (0.618 + 1.07i)19-s + (−2 − 3.46i)23-s + (−2.73 + 4.73i)25-s − 5.52·27-s − 4.47·29-s + (1.23 − 2.14i)31-s + (3.99 + 6.92i)33-s + (2.23 + 3.87i)37-s + ⋯
L(s)  = 1  + (−0.356 + 0.618i)3-s + (−0.723 − 1.25i)5-s + (0.245 + 0.424i)9-s + (0.975 − 1.68i)11-s + 0.211·13-s + 1.03·15-s + (−0.542 + 0.939i)17-s + (0.141 + 0.245i)19-s + (−0.417 − 0.722i)23-s + (−0.547 + 0.947i)25-s − 1.06·27-s − 0.830·29-s + (0.222 − 0.384i)31-s + (0.696 + 1.20i)33-s + (0.367 + 0.636i)37-s + ⋯

Functional equation

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

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

\(L(1)\) \(\approx\) \(0.8525902898\)
\(L(\frac12)\) \(\approx\) \(0.8525902898\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.61 + 2.80i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3.23 + 5.60i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 0.763T + 13T^{2} \)
17 \( 1 + (2.23 - 3.87i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.618 - 1.07i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (2 + 3.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (-1.23 + 2.14i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.23 - 3.87i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 8.47T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + (5.23 + 9.06i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-5 + 8.66i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.61 + 7.99i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.61 + 9.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2 - 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.94T + 71T^{2} \)
73 \( 1 + (-1.47 + 2.54i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.47 + 11.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.23T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.999778912438799180014019300453, −8.452840322309361131379856582175, −7.940440394796006830922381207770, −6.58073372645822217537051123570, −5.77691905411610541340483041319, −4.94107167724214805076295959782, −4.08446188428997963549101603103, −3.56913284623922601239430305720, −1.68243890793245558774064155403, −0.36759290674107369404878430666, 1.44409773218803673747352396315, 2.64483622295355450505381806943, 3.79698702164160326785658745577, 4.47663708869368072595324037714, 5.86523092570553318608334144575, 6.75636152287216025516398345189, 7.21657162893771429831772512587, 7.57982679582956193147689375080, 9.050218095170145378787393549276, 9.648855976807499816326351007524

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