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Properties

Label 2-1568-7.2-c1-0-6
Degree $2$
Conductor $1568$
Sign $-0.991 - 0.126i$
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  + (1.32 + 2.29i)3-s + (−1.5 + 2.59i)5-s + (−2 + 3.46i)9-s + (1.32 + 2.29i)11-s + 4·13-s − 7.93·15-s + (0.5 + 0.866i)17-s + (−3.96 + 6.87i)19-s + (1.32 − 2.29i)23-s + (−2 − 3.46i)25-s − 2.64·27-s − 4·29-s + (−1.32 − 2.29i)31-s + (−3.5 + 6.06i)33-s + (2.5 − 4.33i)37-s + ⋯
L(s)  = 1  + (0.763 + 1.32i)3-s + (−0.670 + 1.16i)5-s + (−0.666 + 1.15i)9-s + (0.398 + 0.690i)11-s + 1.10·13-s − 2.04·15-s + (0.121 + 0.210i)17-s + (−0.910 + 1.57i)19-s + (0.275 − 0.477i)23-s + (−0.400 − 0.692i)25-s − 0.509·27-s − 0.742·29-s + (−0.237 − 0.411i)31-s + (−0.609 + 1.05i)33-s + (0.410 − 0.711i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.991 - 0.126i$
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: \((2,\ 1568,\ (\ :1/2),\ -0.991 - 0.126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.824372436\)
\(L(\frac12)\) \(\approx\) \(1.824372436\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 2.29i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (1.5 - 2.59i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.32 - 2.29i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.96 - 6.87i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.32 + 2.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 + (1.32 + 2.29i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + (-1.32 + 2.29i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (3.5 + 6.06i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-1.32 - 2.29i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.5 - 4.33i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.32 + 2.29i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (4.5 + 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.32 + 2.29i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.5T + 83T^{2} \)
89 \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8T + 97T^{2} \)
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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

−9.853145867680343892973540485951, −9.073753578069126770416680348110, −8.312658891369335587399929008420, −7.59283792115163947673173264560, −6.61453417777548302806419253111, −5.73892888823593879777510622304, −4.35701114109706879431617127136, −3.83509034834283238655207714865, −3.25359247118708886094269284948, −2.03253600871557335636478912033, 0.67599148624671097610713890734, 1.51608968824669794351611022600, 2.82723075995173206322032745485, 3.81847521224688651091644886398, 4.81219761448339257871423198158, 5.96796962843821459032413341514, 6.78815794517940909190381786770, 7.58034303409810061179954527209, 8.369866771477828494518843932373, 8.775151021663160709486509208936

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