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Properties

Label 2-2340-2340.259-c0-0-5
Degree $2$
Conductor $2340$
Sign $0.766 + 0.642i$
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  + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 0.999·8-s + 9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s − 19-s + (−0.499 + 0.866i)20-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)2-s + 3-s + (−0.499 − 0.866i)4-s + (−0.5 − 0.866i)5-s + (−0.5 + 0.866i)6-s + 0.999·8-s + 9-s + 0.999·10-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)12-s + (−0.5 − 0.866i)13-s + (−0.5 − 0.866i)15-s + (−0.5 + 0.866i)16-s + (−0.5 + 0.866i)18-s − 19-s + (−0.499 + 0.866i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(1.16781\)
Root analytic conductor: \(1.08065\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (259, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :0),\ 0.766 + 0.642i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.050484585\)
\(L(\frac12)\) \(\approx\) \(1.050484585\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 - T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + T + T^{2} \)
41 \( 1 + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 - 2T + T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−8.661055699589130180535151441461, −8.300756641870889131911582708735, −8.015559584416281026620576784725, −6.83542237563253147449215930732, −6.23477615010856153496479284269, −5.03868742953076065479452388364, −4.41241174789999351779192572766, −3.49053203116501723776213759790, −2.12569077275718931684522299537, −0.75725747379875438284348863245, 1.81343786534014853727541099872, 2.32850402317742262407898713786, 3.57166649761305001521953794087, 3.92999079637170995273045792389, 4.88013061056666931797370596065, 6.66136255041799322787354412926, 7.13471435504345859573964188907, 7.893741555564120625044813261275, 8.542610144520106586035921637335, 9.447269725741462308409749414728

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