Properties

Label 2-1840-1.1-c1-0-21
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  + 2.56·3-s + 5-s + 0.819·7-s + 3.56·9-s − 5.38·11-s + 2.46·13-s + 2.56·15-s + 4.20·17-s + 5.38·19-s + 2.09·21-s + 23-s + 25-s + 1.43·27-s + 2.35·29-s − 6.66·31-s − 13.7·33-s + 0.819·35-s + 7.42·37-s + 6.31·39-s + 5.64·41-s + 1.90·43-s + 3.56·45-s + 9.33·47-s − 6.32·49-s + 10.7·51-s + 13.8·53-s − 5.38·55-s + ⋯
L(s)  = 1  + 1.47·3-s + 0.447·5-s + 0.309·7-s + 1.18·9-s − 1.62·11-s + 0.683·13-s + 0.661·15-s + 1.02·17-s + 1.23·19-s + 0.457·21-s + 0.208·23-s + 0.200·25-s + 0.276·27-s + 0.437·29-s − 1.19·31-s − 2.40·33-s + 0.138·35-s + 1.22·37-s + 1.01·39-s + 0.881·41-s + 0.290·43-s + 0.530·45-s + 1.36·47-s − 0.904·49-s + 1.50·51-s + 1.90·53-s − 0.726·55-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

\(L(1)\) \(\approx\) \(3.303154066\)
\(L(\frac12)\) \(\approx\) \(3.303154066\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.56T + 3T^{2} \)
7 \( 1 - 0.819T + 7T^{2} \)
11 \( 1 + 5.38T + 11T^{2} \)
13 \( 1 - 2.46T + 13T^{2} \)
17 \( 1 - 4.20T + 17T^{2} \)
19 \( 1 - 5.38T + 19T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 6.66T + 31T^{2} \)
37 \( 1 - 7.42T + 37T^{2} \)
41 \( 1 - 5.64T + 41T^{2} \)
43 \( 1 - 1.90T + 43T^{2} \)
47 \( 1 - 9.33T + 47T^{2} \)
53 \( 1 - 13.8T + 53T^{2} \)
59 \( 1 + 5.27T + 59T^{2} \)
61 \( 1 + 8.51T + 61T^{2} \)
67 \( 1 + 4.10T + 67T^{2} \)
71 \( 1 - 11.7T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 + 13.7T + 79T^{2} \)
83 \( 1 + 3.32T + 83T^{2} \)
89 \( 1 + 3.58T + 89T^{2} \)
97 \( 1 + 9.02T + 97T^{2} \)
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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

−9.175437949302342611283303283211, −8.460628203013040514952497019110, −7.66033047081317460720514849764, −7.40792070593035994203281638614, −5.86517684783595147408429558377, −5.27432250908996983794137005941, −4.07566036678389405489124712426, −3.05302530070692717602824717491, −2.53184868439352273767761380699, −1.29240558217508469784683339977, 1.29240558217508469784683339977, 2.53184868439352273767761380699, 3.05302530070692717602824717491, 4.07566036678389405489124712426, 5.27432250908996983794137005941, 5.86517684783595147408429558377, 7.40792070593035994203281638614, 7.66033047081317460720514849764, 8.460628203013040514952497019110, 9.175437949302342611283303283211

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