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Properties

Label 2-2340-13.3-c1-0-7
Degree $2$
Conductor $2340$
Sign $0.477 - 0.878i$
Analytic cond. $18.6849$
Root an. cond. $4.32261$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + (0.5 − 0.866i)7-s + (1.5 + 2.59i)11-s + (1 + 3.46i)13-s + (−1.5 + 2.59i)17-s + (3.5 − 6.06i)19-s + (−1.5 − 2.59i)23-s + 25-s + (1.5 + 2.59i)29-s − 4·31-s + (−0.5 + 0.866i)35-s + (3.5 + 6.06i)37-s + (−4.5 − 7.79i)41-s + (−5.5 + 9.52i)43-s + (3 + 5.19i)49-s + ⋯
L(s)  = 1  − 0.447·5-s + (0.188 − 0.327i)7-s + (0.452 + 0.783i)11-s + (0.277 + 0.960i)13-s + (−0.363 + 0.630i)17-s + (0.802 − 1.39i)19-s + (−0.312 − 0.541i)23-s + 0.200·25-s + (0.278 + 0.482i)29-s − 0.718·31-s + (−0.0845 + 0.146i)35-s + (0.575 + 0.996i)37-s + (−0.702 − 1.21i)41-s + (−0.838 + 1.45i)43-s + (0.428 + 0.742i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.477 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2340\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 13\)
Sign: $0.477 - 0.878i$
Analytic conductor: \(18.6849\)
Root analytic conductor: \(4.32261\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2340} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2340,\ (\ :1/2),\ 0.477 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.509913156\)
\(L(\frac12)\) \(\approx\) \(1.509913156\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
13 \( 1 + (-1 - 3.46i)T \)
good7 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (1.5 - 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.5 - 2.59i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4T + 31T^{2} \)
37 \( 1 + (-3.5 - 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.5 + 7.79i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.5 - 9.52i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.5 - 6.06i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.5 - 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (-7.5 - 12.9i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-3.5 + 6.06i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.045335415708274092669804316075, −8.461652262778567891311077241473, −7.41367355618281310070987899494, −6.93305985241364853137976797836, −6.17360886011429375901506559812, −4.89388173357961108999736896143, −4.38677785551368905432978521657, −3.50178570023117928217447544442, −2.29278209313122599247025798979, −1.15359946678190835221702018690, 0.58893967528297735895479420427, 1.93127771079067251891264808421, 3.28549833200437993623794940665, 3.73908379344676860076380414262, 5.01748372213343689106150026376, 5.68779937876947023732891896898, 6.44494410167033123207761967230, 7.53682526483111914768791557216, 8.037229702763945028520136400748, 8.781573425291594366532997271134

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