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Properties

Label 2-2856-357.254-c0-0-1
Degree $2$
Conductor $2856$
Sign $0.947 - 0.319i$
Analytic cond. $1.42532$
Root an. cond. $1.19387$
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  + (−0.866 + 0.5i)3-s + (−0.707 + 1.22i)5-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.707 − 1.22i)11-s + 13-s − 1.41i·15-s + (0.258 − 0.965i)17-s + (0.5 − 0.866i)19-s − 0.999·21-s + (0.707 − 1.22i)23-s + (−0.499 − 0.866i)25-s + 0.999i·27-s + 1.41·29-s + (0.866 − 0.5i)31-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.707 + 1.22i)5-s + (0.866 + 0.5i)7-s + (0.499 − 0.866i)9-s + (−0.707 − 1.22i)11-s + 13-s − 1.41i·15-s + (0.258 − 0.965i)17-s + (0.5 − 0.866i)19-s − 0.999·21-s + (0.707 − 1.22i)23-s + (−0.499 − 0.866i)25-s + 0.999i·27-s + 1.41·29-s + (0.866 − 0.5i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 - 0.319i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2856\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 17\)
Sign: $0.947 - 0.319i$
Analytic conductor: \(1.42532\)
Root analytic conductor: \(1.19387\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2856} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2856,\ (\ :0),\ 0.947 - 0.319i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9140693350\)
\(L(\frac12)\) \(\approx\) \(0.9140693350\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (-0.258 + 0.965i)T \)
good5 \( 1 + (0.707 - 1.22i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.707 + 1.22i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (1.22 - 0.707i)T + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-1.73 - i)T + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
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

−8.816226023516867900258263570582, −8.353986475353264443206357205675, −7.37372464828601947955951434042, −6.66339501262297345950326925766, −5.95691150463995575344318803111, −5.11450371066332101229643179725, −4.43436296629669348895250682070, −3.24543520960357508388533110072, −2.77502671536084524455265104224, −0.824662457186275872128059312635, 1.17700452929775926458350364842, 1.67937218360366227477477452300, 3.53508302917532419151076616773, 4.55226805240665878657180323909, 4.93872805387823805781737925196, 5.67909866606437561710260601275, 6.76249889898839278475743299116, 7.49476654444483465751468812516, 8.277390435049315984713304896765, 8.413340390124234442137295199636

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