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Properties

Label 2-1568-56.3-c1-0-4
Degree $2$
Conductor $1568$
Sign $-0.732 - 0.681i$
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  + (−1.5 − 2.59i)9-s + (−2 + 3.46i)11-s + (−4.58 + 2.64i)23-s + (2.5 − 4.33i)25-s + 10.5i·29-s + (−9.16 + 5.29i)37-s − 12·43-s + (9.16 + 5.29i)53-s + (2 − 3.46i)67-s + 5.29i·71-s + (−13.7 + 7.93i)79-s + (−4.5 + 7.79i)81-s + 12·99-s + (−10 − 17.3i)107-s + (−9.16 − 5.29i)109-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)9-s + (−0.603 + 1.04i)11-s + (−0.955 + 0.551i)23-s + (0.5 − 0.866i)25-s + 1.96i·29-s + (−1.50 + 0.869i)37-s − 1.82·43-s + (1.25 + 0.726i)53-s + (0.244 − 0.423i)67-s + 0.627i·71-s + (−1.54 + 0.893i)79-s + (−0.5 + 0.866i)81-s + 1.20·99-s + (−0.966 − 1.67i)107-s + (−0.877 − 0.506i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1568\)    =    \(2^{5} \cdot 7^{2}\)
Sign: $-0.732 - 0.681i$
Analytic conductor: \(12.5205\)
Root analytic conductor: \(3.53843\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1568} (815, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1568,\ (\ :1/2),\ -0.732 - 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5494160220\)
\(L(\frac12)\) \(\approx\) \(0.5494160220\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (2 - 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.58 - 2.64i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 10.5iT - 29T^{2} \)
31 \( 1 + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (9.16 - 5.29i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-9.16 - 5.29i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (13.7 - 7.93i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.872642665648951553315461163060, −8.832242278659656635049908830340, −8.302924882467537704016559627662, −7.17933346152783010543411705321, −6.65896354791639757718106562660, −5.57476551835550563598734468394, −4.83429387965558064059254086525, −3.73729637171714726921579559223, −2.81487944975444425011591682469, −1.56112632682335393001121388384, 0.20290717073448252497701460474, 1.98651638712479665177787825033, 2.95960716319372702483521309163, 3.99213168314673067241923837282, 5.16413244200806837360442507183, 5.71003656592718277045397292254, 6.66765093633596034379100611247, 7.73492705159133692780338756440, 8.290579965046777128779886426017, 8.957047473407852491828481398446

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