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Properties

Label 2-1848-1848.1517-c0-0-3
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

\(L(\frac{1}{2})\) \(\approx\) \(2.580452269\)
\(L(\frac12)\) \(\approx\) \(2.580452269\)
\(L(1)\) 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

−9.291181901654063081318183227474, −8.583308995281298606228149787215, −7.53002263750398275768570783735, −7.19951403608889660223405887519, −6.31704726999626991858715248451, −5.67873168332637091217237140718, −4.50868203174925932071374921751, −3.40269854316847308257156299863, −2.91078674448830522328120577419, −1.79855032065337645803465258735, 1.92740938207215443983903201757, 2.42547144954059684996973596836, 3.42796406493232598834209839851, 4.40839346741739135600643622183, 5.36000362075818885679509300705, 5.71734149921308956371119684713, 6.89511247855689996176848927760, 7.897866790270265114871524212137, 8.902440947701213923079245874526, 9.630492210184055345181788837282

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