Properties

Label 2-1840-1.1-c1-0-24
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.68·3-s − 5-s + 4.59·7-s + 4.22·9-s − 5.13·11-s − 1.22·13-s + 2.68·15-s − 4.68·17-s + 4.59·19-s − 12.3·21-s + 23-s + 25-s − 3.28·27-s + 3.37·29-s + 0.777·31-s + 13.7·33-s − 4.59·35-s + 5.81·37-s + 3.28·39-s − 8.50·41-s − 8·43-s − 4.22·45-s + 6.44·47-s + 14.1·49-s + 12.5·51-s − 6·53-s + 5.13·55-s + ⋯
L(s)  = 1  − 1.55·3-s − 0.447·5-s + 1.73·7-s + 1.40·9-s − 1.54·11-s − 0.338·13-s + 0.693·15-s − 1.13·17-s + 1.05·19-s − 2.69·21-s + 0.208·23-s + 0.200·25-s − 0.632·27-s + 0.626·29-s + 0.139·31-s + 2.40·33-s − 0.777·35-s + 0.956·37-s + 0.525·39-s − 1.32·41-s − 1.21·43-s − 0.629·45-s + 0.939·47-s + 2.01·49-s + 1.76·51-s − 0.824·53-s + 0.691·55-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: \(1\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.68T + 3T^{2} \)
7 \( 1 - 4.59T + 7T^{2} \)
11 \( 1 + 5.13T + 11T^{2} \)
13 \( 1 + 1.22T + 13T^{2} \)
17 \( 1 + 4.68T + 17T^{2} \)
19 \( 1 - 4.59T + 19T^{2} \)
29 \( 1 - 3.37T + 29T^{2} \)
31 \( 1 - 0.777T + 31T^{2} \)
37 \( 1 - 5.81T + 37T^{2} \)
41 \( 1 + 8.50T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 - 6.44T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 9.37T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 15.6T + 67T^{2} \)
71 \( 1 + 1.31T + 71T^{2} \)
73 \( 1 + 4.44T + 73T^{2} \)
79 \( 1 - 4.88T + 79T^{2} \)
83 \( 1 - 3.81T + 83T^{2} \)
89 \( 1 - 8.93T + 89T^{2} \)
97 \( 1 + 18.0T + 97T^{2} \)
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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

−8.671000380389074055058138627513, −7.902244953715744039878273262532, −7.34372772175870721098056832710, −6.39155683924310895797136880991, −5.22322293436153291608865640477, −5.07185923159146725553510670007, −4.31951891229441644583173274331, −2.66822144317941243329337021161, −1.35504653420528706663932098210, 0, 1.35504653420528706663932098210, 2.66822144317941243329337021161, 4.31951891229441644583173274331, 5.07185923159146725553510670007, 5.22322293436153291608865640477, 6.39155683924310895797136880991, 7.34372772175870721098056832710, 7.902244953715744039878273262532, 8.671000380389074055058138627513

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