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Properties

Label 2-1840-460.459-c1-0-33
Degree $2$
Conductor $1840$
Sign $0.114 + 0.993i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
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  − 3.28·3-s + (−1.54 − 1.62i)5-s − 2.73i·7-s + 7.80·9-s + 2.28·11-s − 4.92i·13-s + (5.06 + 5.32i)15-s + 5.97·17-s − 0.377·19-s + 8.97i·21-s + (−0.582 + 4.76i)23-s + (−0.249 + 4.99i)25-s − 15.7·27-s + 3.70·29-s + 2.89i·31-s + ⋯
L(s)  = 1  − 1.89·3-s + (−0.689 − 0.724i)5-s − 1.03i·7-s + 2.60·9-s + 0.689·11-s − 1.36i·13-s + (1.30 + 1.37i)15-s + 1.44·17-s − 0.0866·19-s + 1.95i·21-s + (−0.121 + 0.992i)23-s + (−0.0498 + 0.998i)25-s − 3.03·27-s + 0.687·29-s + 0.519i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8718351961\)
\(L(\frac12)\) \(\approx\) \(0.8718351961\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (1.54 + 1.62i)T \)
23 \( 1 + (0.582 - 4.76i)T \)
good3 \( 1 + 3.28T + 3T^{2} \)
7 \( 1 + 2.73iT - 7T^{2} \)
11 \( 1 - 2.28T + 11T^{2} \)
13 \( 1 + 4.92iT - 13T^{2} \)
17 \( 1 - 5.97T + 17T^{2} \)
19 \( 1 + 0.377T + 19T^{2} \)
29 \( 1 - 3.70T + 29T^{2} \)
31 \( 1 - 2.89iT - 31T^{2} \)
37 \( 1 - 7.23T + 37T^{2} \)
41 \( 1 - 0.451T + 41T^{2} \)
43 \( 1 - 0.359iT - 43T^{2} \)
47 \( 1 - 12.1T + 47T^{2} \)
53 \( 1 - 9.69T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 6.73iT - 61T^{2} \)
67 \( 1 + 7.06iT - 67T^{2} \)
71 \( 1 + 7.34iT - 71T^{2} \)
73 \( 1 - 1.24iT - 73T^{2} \)
79 \( 1 - 13.6T + 79T^{2} \)
83 \( 1 - 15.1iT - 83T^{2} \)
89 \( 1 + 12.1iT - 89T^{2} \)
97 \( 1 + 4.34T + 97T^{2} \)
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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

−9.282332741529939230218164343700, −7.86893589486134644722916692903, −7.52806807425904415944738256430, −6.60630800734063300847288436838, −5.65696973162364430613297672885, −5.17406409921121198364312565713, −4.22584284436331959307397840324, −3.56925306534337150369362537837, −1.12567485029633983992345793420, −0.71742853301410722833371078162, 0.925038505776953209271499428601, 2.39045370491749922129670932208, 3.94072917790433806890727694023, 4.52541390110329045306059643072, 5.64119278829290908586293472311, 6.17023670575000439811715927829, 6.84852540953571323257950424729, 7.53628926268574041498004327203, 8.699257318674947658535410929248, 9.663084488414993370969107909159

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