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Properties

Label 2-2340-260.123-c0-0-2
Degree $2$
Conductor $2340$
Sign $0.439 + 0.898i$
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 + (−0.866 + 0.499i)10-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)16-s + (0.133 − 0.5i)17-s + 0.999·20-s + (−0.499 − 0.866i)25-s + (−0.499 − 0.866i)26-s + (1.5 + 0.866i)29-s + (0.866 − 0.499i)32-s + (−0.366 + 0.366i)34-s + (0.5 − 0.866i)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 + (−0.866 + 0.499i)10-s + (0.866 + 0.5i)13-s + (−0.5 + 0.866i)16-s + (0.133 − 0.5i)17-s + 0.999·20-s + (−0.499 − 0.866i)25-s + (−0.499 − 0.866i)26-s + (1.5 + 0.866i)29-s + (0.866 − 0.499i)32-s + (−0.366 + 0.366i)34-s + (0.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9182392786\)
\(L(\frac12)\) \(\approx\) \(0.9182392786\)
\(L(1)\) 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 \( 1 + (-0.866 - 0.5i)T \)
good7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.866 + 0.5i)T^{2} \)
17 \( 1 + (-0.133 + 0.5i)T + (-0.866 - 0.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + iT^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.133 + 0.5i)T + (-0.866 + 0.5i)T^{2} \)
43 \( 1 + (-0.866 - 0.5i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (1.36 + 1.36i)T + iT^{2} \)
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 \( 1 + 1.73iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.366 - 1.36i)T + (-0.866 + 0.5i)T^{2} \)
97 \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \)
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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

−9.041084053171913995291354546042, −8.518602664697166949122448618756, −7.77371228048910392158194726840, −6.78650273521774607495531883758, −6.09031183282062896534119948187, −4.99063329368132907593475931416, −4.10410455283999743017180934824, −3.05544443302310310816262444219, −1.94689161035541738097130901430, −0.993157282828790904401200618972, 1.27628150023392455655018465167, 2.44063983252321903168013221107, 3.36952763789750718959975770984, 4.73209055226615999281784466579, 5.80334396138848087907223232122, 6.31527067758583384913734307912, 6.92099433712526511661440640688, 7.993086798339881094004979509214, 8.322723836499336482754905793415, 9.392850692525734540314286346993

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