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Properties

Label 2-2340-260.223-c0-0-1
Degree $2$
Conductor $2340$
Sign $0.998 + 0.0557i$
Analytic cond. $1.16781$
Root an. cond. $1.08065$
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 + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999i·8-s + 0.999i·10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (1.86 − 0.5i)17-s + (−0.499 − 0.866i)20-s + (−0.499 − 0.866i)25-s + (−0.866 − 0.499i)26-s + (−0.866 + 0.5i)29-s + (0.866 + 0.499i)32-s + (−1.36 + 1.36i)34-s + (0.866 + 1.5i)37-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (0.5 − 0.866i)5-s + 0.999i·8-s + 0.999i·10-s + (0.5 + 0.866i)13-s + (−0.5 − 0.866i)16-s + (1.86 − 0.5i)17-s + (−0.499 − 0.866i)20-s + (−0.499 − 0.866i)25-s + (−0.866 − 0.499i)26-s + (−0.866 + 0.5i)29-s + (0.866 + 0.499i)32-s + (−1.36 + 1.36i)34-s + (0.866 + 1.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0557i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.998 + 0.0557i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1783, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ 0.998 + 0.0557i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9385491084\)
\(L(\frac12)\) \(\approx\) \(0.9385491084\)
\(L(1)\) 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 \( 1 \)
5 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.866 + 0.5i)T^{2} \)
17 \( 1 + (-1.86 + 0.5i)T + (0.866 - 0.5i)T^{2} \)
19 \( 1 + (-0.866 + 0.5i)T^{2} \)
23 \( 1 + (-0.866 - 0.5i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 + (0.5 + 0.133i)T + (0.866 + 0.5i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-0.366 - 0.366i)T + iT^{2} \)
59 \( 1 + (0.866 - 0.5i)T^{2} \)
61 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T^{2} \)
73 \( 1 + iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (1.36 + 0.366i)T + (0.866 + 0.5i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−9.167133059553393619732447419790, −8.420921873823650466644103825660, −7.81079504183611018342282617963, −6.92249881622708960189258484768, −6.09446160193517165910735243330, −5.40331889256978653395252865663, −4.68458995031354895229339435588, −3.33958476469430365857257331550, −1.94403699290133779680436145252, −1.07915593727502063334740732845, 1.20707250774332776105861197395, 2.36933693791612729874955773674, 3.25273887595260172844971689944, 3.90957038239880204748714457849, 5.57839964075950981265432986474, 6.06000156412875539359960910176, 7.15833806368652436414800583779, 7.71553840784388557971849401970, 8.394084062110201801378161233282, 9.398923164113387056969051185028

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