Properties

Label 2-1840-5.4-c1-0-9
Degree $2$
Conductor $1840$
Sign $-0.703 + 0.711i$
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  + 3.21i·3-s + (1.59 + 1.57i)5-s + 2.43i·7-s − 7.36·9-s + 0.884·11-s − 5.10i·13-s + (−5.06 + 5.11i)15-s + 0.366i·17-s − 2.79·19-s − 7.82·21-s + i·23-s + (0.0570 + 4.99i)25-s − 14.0i·27-s − 8.02·29-s − 7.24·31-s + ⋯
L(s)  = 1  + 1.85i·3-s + (0.711 + 0.703i)5-s + 0.919i·7-s − 2.45·9-s + 0.266·11-s − 1.41i·13-s + (−1.30 + 1.32i)15-s + 0.0889i·17-s − 0.642·19-s − 1.70·21-s + 0.208i·23-s + (0.0114 + 0.999i)25-s − 2.70i·27-s − 1.49·29-s − 1.30·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.711i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.703 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.703 + 0.711i$
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: \((2,\ 1840,\ (\ :1/2),\ -0.703 + 0.711i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.172824141\)
\(L(\frac12)\) \(\approx\) \(1.172824141\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.59 - 1.57i)T \)
23 \( 1 - iT \)
good3 \( 1 - 3.21iT - 3T^{2} \)
7 \( 1 - 2.43iT - 7T^{2} \)
11 \( 1 - 0.884T + 11T^{2} \)
13 \( 1 + 5.10iT - 13T^{2} \)
17 \( 1 - 0.366iT - 17T^{2} \)
19 \( 1 + 2.79T + 19T^{2} \)
29 \( 1 + 8.02T + 29T^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 + 3.10iT - 37T^{2} \)
41 \( 1 + 3.47T + 41T^{2} \)
43 \( 1 - 8.56iT - 43T^{2} \)
47 \( 1 + 5.25iT - 47T^{2} \)
53 \( 1 - 11.6iT - 53T^{2} \)
59 \( 1 + 9.33T + 59T^{2} \)
61 \( 1 - 5.46T + 61T^{2} \)
67 \( 1 + 1.49iT - 67T^{2} \)
71 \( 1 + 8.29T + 71T^{2} \)
73 \( 1 - 10.2iT - 73T^{2} \)
79 \( 1 - 6.06T + 79T^{2} \)
83 \( 1 - 16.2iT - 83T^{2} \)
89 \( 1 - 17.6T + 89T^{2} \)
97 \( 1 + 6.55iT - 97T^{2} \)
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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

−9.659387698857827672790373722351, −9.225449230938095772859379710544, −8.527174083605075344397303007230, −7.48247394985559869317796087503, −6.06807617867468666561513532998, −5.67200257896097315237787554425, −4.99819856183235241257623790245, −3.80469987113897297498397114526, −3.14335522548054440541972421044, −2.23187766836765641796634372713, 0.39842541223044207724829568557, 1.68320684064780484707927325645, 2.02203744558350765694063104444, 3.61841177372382279159838317234, 4.77219070911321389938608452980, 5.83600503173571264239957403292, 6.50790061547193809816460113279, 7.14015405656532485529070995803, 7.76617794598888727014826182935, 8.819396588945359757174766131381

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