Properties

Label 2-1840-5.4-c1-0-32
Degree $2$
Conductor $1840$
Sign $0.574 + 0.818i$
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  − 0.706i·3-s + (−1.83 + 1.28i)5-s − 2.40i·7-s + 2.50·9-s − 4.95·11-s + 4.40i·13-s + (0.907 + 1.29i)15-s + 0.200i·17-s + 6.65·19-s − 1.70·21-s i·23-s + (1.70 − 4.70i)25-s − 3.88i·27-s − 1.67·29-s + 1.82·31-s + ⋯
L(s)  = 1  − 0.407i·3-s + (−0.818 + 0.574i)5-s − 0.910i·7-s + 0.833·9-s − 1.49·11-s + 1.22i·13-s + (0.234 + 0.333i)15-s + 0.0487i·17-s + 1.52·19-s − 0.371·21-s − 0.208i·23-s + (0.340 − 0.940i)25-s − 0.748i·27-s − 0.311·29-s + 0.327·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ 0.574 + 0.818i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.333035953\)
\(L(\frac12)\) \(\approx\) \(1.333035953\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 - 1.28i)T \)
23 \( 1 + iT \)
good3 \( 1 + 0.706iT - 3T^{2} \)
7 \( 1 + 2.40iT - 7T^{2} \)
11 \( 1 + 4.95T + 11T^{2} \)
13 \( 1 - 4.40iT - 13T^{2} \)
17 \( 1 - 0.200iT - 17T^{2} \)
19 \( 1 - 6.65T + 19T^{2} \)
29 \( 1 + 1.67T + 29T^{2} \)
31 \( 1 - 1.82T + 31T^{2} \)
37 \( 1 + 8.47iT - 37T^{2} \)
41 \( 1 - 11.0T + 41T^{2} \)
43 \( 1 - 2.06iT - 43T^{2} \)
47 \( 1 + 0.505iT - 47T^{2} \)
53 \( 1 + 5.84iT - 53T^{2} \)
59 \( 1 + 4.41T + 59T^{2} \)
61 \( 1 + 6.67T + 61T^{2} \)
67 \( 1 + 12.6iT - 67T^{2} \)
71 \( 1 + 3.61T + 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 - 13.3iT - 83T^{2} \)
89 \( 1 - 5.04T + 89T^{2} \)
97 \( 1 - 3.79iT - 97T^{2} \)
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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

−9.212645972975801342738712670468, −7.907623217893294508357001769496, −7.52633114563504091952839027601, −7.07336543859083890002469162182, −6.11355484440015075603194295802, −4.85521235091434571253152919696, −4.15353299621162149152189370360, −3.23365339621811788380737729166, −2.07759460230071546164048947570, −0.63024403685723626090783386600, 1.00087277978112737784433654083, 2.66177337767101645815840262254, 3.42267977710685085425275680543, 4.61522064593270324970723902258, 5.22584378834960446550056392959, 5.85994305069937356146169026049, 7.45913477315150242467658577691, 7.67534320532134209615980248531, 8.560993920467997011057584580179, 9.371393982631692526569961031433

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