Properties

Label 2-1840-5.4-c1-0-47
Degree $2$
Conductor $1840$
Sign $-0.217 + 0.976i$
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  − 0.296i·3-s + (−2.18 − 0.485i)5-s − 3.46i·7-s + 2.91·9-s + 3.11·11-s − 4.60i·13-s + (−0.144 + 0.647i)15-s + 5.49i·17-s + 4.48·19-s − 1.02·21-s + i·23-s + (4.52 + 2.11i)25-s − 1.75i·27-s − 9.19·29-s + 5.89·31-s + ⋯
L(s)  = 1  − 0.171i·3-s + (−0.976 − 0.217i)5-s − 1.30i·7-s + 0.970·9-s + 0.938·11-s − 1.27i·13-s + (−0.0372 + 0.167i)15-s + 1.33i·17-s + 1.02·19-s − 0.224·21-s + 0.208i·23-s + (0.905 + 0.423i)25-s − 0.337i·27-s − 1.70·29-s + 1.05·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.217 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.217 + 0.976i$
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: \((2,\ 1840,\ (\ :1/2),\ -0.217 + 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.490979678\)
\(L(\frac12)\) \(\approx\) \(1.490979678\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.18 + 0.485i)T \)
23 \( 1 - iT \)
good3 \( 1 + 0.296iT - 3T^{2} \)
7 \( 1 + 3.46iT - 7T^{2} \)
11 \( 1 - 3.11T + 11T^{2} \)
13 \( 1 + 4.60iT - 13T^{2} \)
17 \( 1 - 5.49iT - 17T^{2} \)
19 \( 1 - 4.48T + 19T^{2} \)
29 \( 1 + 9.19T + 29T^{2} \)
31 \( 1 - 5.89T + 31T^{2} \)
37 \( 1 + 6.95iT - 37T^{2} \)
41 \( 1 + 9.03T + 41T^{2} \)
43 \( 1 + 5.55iT - 43T^{2} \)
47 \( 1 + 5.48iT - 47T^{2} \)
53 \( 1 - 2.74iT - 53T^{2} \)
59 \( 1 - 9.33T + 59T^{2} \)
61 \( 1 + 1.40T + 61T^{2} \)
67 \( 1 + 3.49iT - 67T^{2} \)
71 \( 1 + 4.28T + 71T^{2} \)
73 \( 1 + 4.92iT - 73T^{2} \)
79 \( 1 - 2.12T + 79T^{2} \)
83 \( 1 + 16.0iT - 83T^{2} \)
89 \( 1 + 11.9T + 89T^{2} \)
97 \( 1 + 4.37iT - 97T^{2} \)
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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

−8.927822559916527814166751607744, −8.048145675423304577432891071410, −7.40013880937064606658934484555, −6.99312195924544446883783857900, −5.84079981998491221608495349779, −4.74776820867859524685835609992, −3.75749175554516146351978255034, −3.59327870236263170437813040986, −1.57972855240767754393617180671, −0.63108596129677924280752603424, 1.37672700138888019240358822040, 2.66817006527896983852478602119, 3.68114699036703668901939043811, 4.51992178302514849246506552279, 5.26952331574476155395705473585, 6.54744523872152637738964872851, 7.00296361274197920577136524972, 7.88705046831548011577239092348, 8.844909908661519344177881618936, 9.397863048522847434290000823974

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