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Properties

Label 2-1848-1848.1517-c0-0-1
Degree $2$
Conductor $1848$
Sign $-0.922 + 0.386i$
Analytic cond. $0.922272$
Root an. cond. $0.960350$
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)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.866 − 0.499i)12-s + 0.999i·14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.866 − 0.5i)5-s + 0.999·6-s + (−0.5 − 0.866i)7-s − 0.999i·8-s + (0.499 − 0.866i)9-s + (0.499 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.866 − 0.499i)12-s + 0.999i·14-s + 0.999·15-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.499i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1848 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.922 + 0.386i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1848\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.922 + 0.386i$
Analytic conductor: \(0.922272\)
Root analytic conductor: \(0.960350\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1848} (1517, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1848,\ (\ :0),\ -0.922 + 0.386i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2134327401\)
\(L(\frac12)\) \(\approx\) \(0.2134327401\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + iT - T^{2} \)
31 \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + 1.73T + T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + 1.73iT - T^{2} \)
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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

−9.280386506405227570838658493405, −8.486615967376488092445331597255, −7.46400349216315738246037768643, −6.94332052049227287562971618158, −6.08631117547424184394253699853, −4.67955904025606569903355095420, −4.01228487807515736745203268338, −3.38699441352033818417387591852, −1.53633046803468696684619601117, −0.25484365420239237339543136201, 1.44025138369517005013955168426, 2.76226419758507908969048800659, 4.04662844674382286034275585461, 5.42044717464037627872122352158, 5.92153321747872097565987424365, 6.81114998332857498268084467388, 7.25347504403494627566007792411, 8.135887683812036000474829854154, 8.927097628422380463471715586079, 9.638087923866284070996802075900

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