Properties

Degree $2$
Conductor $1568$
Sign $0.968 + 0.250i$
Motivic weight $1$
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: $0.968 + 0.250i$
Motivic weight: \(1\)
Character: $\chi_{1568} (1537, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ 0.968 + 0.250i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5446436649\)
\(L(\frac12)\) \(\approx\) \(0.5446436649\)
\(L(\frac{3}{2})\) 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

−9.727792665595658806170567889271, −8.775081635232506789533455307331, −7.988425110978159178067378461054, −6.77871022911992178770189477970, −5.90385733966935066186706146635, −5.00433023383203266845725630757, −4.52625896132045319689355619955, −3.81364795434979051880920249280, −2.44068474115014444914559846124, −0.31518776568869975407091987471, 0.881876731799498281997042878081, 2.29446371429252660959904335150, 3.05334201112883358038135800779, 4.92744113464667765127445345725, 5.34091668004399337499352300294, 6.55443480069745489458794101045, 6.95518031472263706758640763724, 7.61783212881687598754646010466, 8.325713162145498666459556249370, 9.464489960646440472347023071574

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