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Properties

Label 2-2808-104.51-c0-0-5
Degree $2$
Conductor $2808$
Sign $-0.707 + 0.707i$
Analytic cond. $1.40137$
Root an. cond. $1.18379$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s − 0.517·5-s + (−0.707 − 0.707i)8-s + (0.499 + 0.133i)10-s − 1.41i·11-s − i·13-s + (0.500 + 0.866i)16-s + (−0.448 − 0.258i)20-s + (−0.366 + 1.36i)22-s − 0.732·25-s + (−0.258 + 0.965i)26-s + (−0.258 − 0.965i)32-s + (0.366 + 0.366i)40-s + 1.41i·41-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s − 0.517·5-s + (−0.707 − 0.707i)8-s + (0.499 + 0.133i)10-s − 1.41i·11-s − i·13-s + (0.500 + 0.866i)16-s + (−0.448 − 0.258i)20-s + (−0.366 + 1.36i)22-s − 0.732·25-s + (−0.258 + 0.965i)26-s + (−0.258 − 0.965i)32-s + (0.366 + 0.366i)40-s + 1.41i·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2808\)    =    \(2^{3} \cdot 3^{3} \cdot 13\)
Sign: $-0.707 + 0.707i$
Analytic conductor: \(1.40137\)
Root analytic conductor: \(1.18379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2808} (2755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2808,\ (\ :0),\ -0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4379435628\)
\(L(\frac12)\) \(\approx\) \(0.4379435628\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 + 0.258i)T \)
3 \( 1 \)
13 \( 1 + iT \)
good5 \( 1 + 0.517T + T^{2} \)
7 \( 1 + T^{2} \)
11 \( 1 + 1.41iT - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + 1.93T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + 0.517iT - T^{2} \)
61 \( 1 + iT - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - 0.517T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + 2iT - T^{2} \)
83 \( 1 + 1.93iT - T^{2} \)
89 \( 1 + 0.517iT - T^{2} \)
97 \( 1 - T^{2} \)
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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

−8.429814449050923917025900104466, −8.199244924915525586601679884313, −7.50650883930940365179253509249, −6.45825309892905759833878622637, −5.90328116194096114310611705525, −4.76904950529658229467527293226, −3.36046915001331098781579109507, −3.18354007348315695390409528052, −1.70597996377480035149156703898, −0.37573271514717903146975070029, 1.56540321045556583297649665411, 2.36442206181223786544943220764, 3.69372141021466191245798097654, 4.63043696915266080993265344278, 5.50769672886503043122183789893, 6.69134494004178227279862770094, 6.96090587824462308778936078069, 7.85528876826953180953670612233, 8.403894440026969263263566026821, 9.408727243425443455294109681400

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