Properties

Label 2-1840-1.1-c1-0-8
Degree $2$
Conductor $1840$
Sign $1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.878·3-s + 5-s − 0.121·7-s − 2.22·9-s − 2.87·11-s + 5.22·13-s − 0.878·15-s − 2.22·17-s − 1.22·19-s + 0.106·21-s − 23-s + 25-s + 4.59·27-s + 9.34·29-s + 2.12·31-s + 2.52·33-s − 0.121·35-s + 5.59·37-s − 4.59·39-s + 8.22·41-s − 8·43-s − 2.22·45-s + 10.4·47-s − 6.98·49-s + 1.95·51-s − 3.59·53-s − 2.87·55-s + ⋯
L(s)  = 1  − 0.507·3-s + 0.447·5-s − 0.0459·7-s − 0.742·9-s − 0.867·11-s + 1.45·13-s − 0.226·15-s − 0.540·17-s − 0.281·19-s + 0.0232·21-s − 0.208·23-s + 0.200·25-s + 0.883·27-s + 1.73·29-s + 0.381·31-s + 0.440·33-s − 0.0205·35-s + 0.919·37-s − 0.735·39-s + 1.28·41-s − 1.21·43-s − 0.332·45-s + 1.52·47-s − 0.997·49-s + 0.274·51-s − 0.493·53-s − 0.388·55-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.388851670\)
\(L(\frac12)\) \(\approx\) \(1.388851670\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 + T \)
good3 \( 1 + 0.878T + 3T^{2} \)
7 \( 1 + 0.121T + 7T^{2} \)
11 \( 1 + 2.87T + 11T^{2} \)
13 \( 1 - 5.22T + 13T^{2} \)
17 \( 1 + 2.22T + 17T^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
29 \( 1 - 9.34T + 29T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 - 5.59T + 37T^{2} \)
41 \( 1 - 8.22T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 10.4T + 47T^{2} \)
53 \( 1 + 3.59T + 53T^{2} \)
59 \( 1 - 0.650T + 59T^{2} \)
61 \( 1 + 7.33T + 61T^{2} \)
67 \( 1 + 5.59T + 67T^{2} \)
71 \( 1 - 13.9T + 71T^{2} \)
73 \( 1 - 12.9T + 73T^{2} \)
79 \( 1 - 3.51T + 79T^{2} \)
83 \( 1 - 11.1T + 83T^{2} \)
89 \( 1 + 0.486T + 89T^{2} \)
97 \( 1 - 0.635T + 97T^{2} \)
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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

−9.174037408893991162217636690479, −8.454063763070601702592249766948, −7.85334589747077589175542377544, −6.44864329636794283897094501435, −6.22084020494635736758782841765, −5.28124812088267100563230489234, −4.46105389608642678139232982384, −3.22545358741562915891247383788, −2.30667044217186402317908011948, −0.819094369520520870719249001010, 0.819094369520520870719249001010, 2.30667044217186402317908011948, 3.22545358741562915891247383788, 4.46105389608642678139232982384, 5.28124812088267100563230489234, 6.22084020494635736758782841765, 6.44864329636794283897094501435, 7.85334589747077589175542377544, 8.454063763070601702592249766948, 9.174037408893991162217636690479

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