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Properties

Label 2-2340-260.159-c0-0-3
Degree $2$
Conductor $2340$
Sign $-0.702 + 0.711i$
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 + (−0.866 − 0.5i)5-s − 0.999i·8-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.499i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 0.499i)26-s + (−0.866 − 1.5i)29-s + (−0.866 − 0.499i)32-s − 0.999·34-s + (1.5 − 0.866i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + (−0.866 − 0.5i)5-s − 0.999i·8-s − 0.999·10-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.866 + 0.499i)20-s + (0.499 + 0.866i)25-s + (−0.866 − 0.499i)26-s + (−0.866 − 1.5i)29-s + (−0.866 − 0.499i)32-s − 0.999·34-s + (1.5 − 0.866i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $-0.702 + 0.711i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (1459, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ -0.702 + 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.435340173\)
\(L(\frac12)\) \(\approx\) \(1.435340173\)
\(L(1)\) 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 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + iT - T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + 1.73iT - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + (-0.5 + 0.866i)T^{2} \)
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

−9.090324439474516032468972809786, −7.86528116982065478189834930068, −7.47369468307186785712034245211, −6.34963665988411493588594886189, −5.59335386538048255474220839280, −4.65146067074393548277043764824, −4.18414491534423442540038253087, −3.13764524901342363360085729479, −2.25533035638406032652542607572, −0.71044187085812859597763986823, 2.06960231900078257004042246077, 3.05446947225381177732920498698, 4.01153916361877504613798231799, 4.51017727356715657098780451605, 5.53178174649521614354643157240, 6.48977830936366570058823423273, 7.06564944371412836476142431046, 7.65444359298530889207378888665, 8.538742383182854866424691149897, 9.199451983604331789357226145990

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