Properties

Label 2-1840-1.1-c1-0-16
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 − 1.56·7-s + 3.56·9-s + 2·11-s + 0.561·13-s − 2.56·15-s + 5.56·17-s + 2·19-s − 4·21-s + 23-s + 25-s + 1.43·27-s + 0.123·29-s + 8.12·31-s + 5.12·33-s + 1.56·35-s − 3.56·37-s + 1.43·39-s − 4.12·41-s + 10.2·43-s − 3.56·45-s − 3.68·47-s − 4.56·49-s + 14.2·51-s + 4.43·53-s − 2·55-s + ⋯
L(s)  = 1  + 1.47·3-s − 0.447·5-s − 0.590·7-s + 1.18·9-s + 0.603·11-s + 0.155·13-s − 0.661·15-s + 1.34·17-s + 0.458·19-s − 0.872·21-s + 0.208·23-s + 0.200·25-s + 0.276·27-s + 0.0228·29-s + 1.45·31-s + 0.891·33-s + 0.263·35-s − 0.585·37-s + 0.230·39-s − 0.643·41-s + 1.56·43-s − 0.530·45-s − 0.537·47-s − 0.651·49-s + 1.99·51-s + 0.609·53-s − 0.269·55-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

\(L(1)\) \(\approx\) \(2.800056566\)
\(L(\frac12)\) \(\approx\) \(2.800056566\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.56T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 - 0.561T + 13T^{2} \)
17 \( 1 - 5.56T + 17T^{2} \)
19 \( 1 - 2T + 19T^{2} \)
29 \( 1 - 0.123T + 29T^{2} \)
31 \( 1 - 8.12T + 31T^{2} \)
37 \( 1 + 3.56T + 37T^{2} \)
41 \( 1 + 4.12T + 41T^{2} \)
43 \( 1 - 10.2T + 43T^{2} \)
47 \( 1 + 3.68T + 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 - 5.56T + 59T^{2} \)
61 \( 1 + 9.12T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 5T + 71T^{2} \)
73 \( 1 - 3.43T + 73T^{2} \)
79 \( 1 - 9.12T + 79T^{2} \)
83 \( 1 - 4.68T + 83T^{2} \)
89 \( 1 - 8T + 89T^{2} \)
97 \( 1 + 3.12T + 97T^{2} \)
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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

−9.271554208610847512874259367228, −8.385665759754422791383462165258, −7.88294530654859239205658542762, −7.08871209573266585050090630276, −6.23074658062119194078754155496, −5.05115189617461357764447206848, −3.85676555192980723489072007430, −3.37702457629928921206777920413, −2.52501082421246270004418908529, −1.14551146797535563427591780627, 1.14551146797535563427591780627, 2.52501082421246270004418908529, 3.37702457629928921206777920413, 3.85676555192980723489072007430, 5.05115189617461357764447206848, 6.23074658062119194078754155496, 7.08871209573266585050090630276, 7.88294530654859239205658542762, 8.385665759754422791383462165258, 9.271554208610847512874259367228

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