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Properties

Label 2-1568-7.2-c1-0-24
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  + (0.866 + 1.5i)3-s + (0.5 − 0.866i)5-s + (−2.59 − 4.5i)11-s + 1.73·15-s + (2.5 + 4.33i)17-s + (0.866 − 1.5i)19-s + (0.866 − 1.5i)23-s + (2 + 3.46i)25-s + 5.19·27-s + 8·29-s + (−4.33 − 7.5i)31-s + (4.5 − 7.79i)33-s + (2.5 − 4.33i)37-s − 4·41-s + 6.92·43-s + ⋯
L(s)  = 1  + (0.499 + 0.866i)3-s + (0.223 − 0.387i)5-s + (−0.783 − 1.35i)11-s + 0.447·15-s + (0.606 + 1.05i)17-s + (0.198 − 0.344i)19-s + (0.180 − 0.312i)23-s + (0.400 + 0.692i)25-s + 1.00·27-s + 1.48·29-s + (−0.777 − 1.34i)31-s + (0.783 − 1.35i)33-s + (0.410 − 0.711i)37-s − 0.624·41-s + 1.05·43-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

\(L(1)\) \(\approx\) \(2.108591103\)
\(L(\frac12)\) \(\approx\) \(2.108591103\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.866 - 1.5i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.59 + 4.5i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (-2.5 - 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.866 + 1.5i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 8T + 29T^{2} \)
31 \( 1 + (4.33 + 7.5i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 4T + 41T^{2} \)
43 \( 1 - 6.92T + 43T^{2} \)
47 \( 1 + (-4.33 + 7.5i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-0.866 - 1.5i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.06 - 10.5i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 13.8T + 71T^{2} \)
73 \( 1 + (-7.5 - 12.9i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.866 + 1.5i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 + (-3.5 + 6.06i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.380251568553941053932138660275, −8.620436843651568649519290470268, −8.189063541130429382809803990343, −7.05201461201838013158204128878, −5.90872325215972691122225698027, −5.32713345874283682131056235472, −4.23636251984765052577611773705, −3.46934673587628436476836886216, −2.56560258576763863827315466905, −0.889032855598072329013689088529, 1.27272331432836158651040106479, 2.41068043866375839986284515278, 3.02049191802053447086076255861, 4.56402819535260686665057841445, 5.25237957900441100985531377473, 6.50024384791012349933762734222, 7.19076328679475227155764079503, 7.67639059105870311286611434884, 8.479440475398574750344532168809, 9.487912164116779066323402117146

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