Properties

Label 2-2340-260.159-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.9538093771\)
\(L(\frac12)\) \(\approx\) \(0.9538093771\)
\(L(1)\) 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

−9.228902043956974448023595887908, −8.592950710894910851609871975237, −7.60147187138419152512691292028, −7.16911942509595140106013255103, −6.12717119897831650680003560538, −5.68189923823844491408262784235, −4.82577927304522875852214016488, −3.25152875438690647159265353840, −2.37067363576697715237417509971, −1.21645416768467545754152740481, 1.05554800917851117845016537503, 2.13562098277091642969601317697, 2.94357341823626712641425667179, 4.21273960370464163810336898745, 5.04071965813992517362004318794, 6.19450713636957629346343554878, 6.77927983456496355626659899432, 7.82782100681435645854092285493, 8.387844353644418589844820426383, 9.269104641270713047944104754465

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