Properties

Label 2-1840-1.1-c1-0-23
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  − 1.43·3-s − 5-s − 3.08·7-s − 0.950·9-s + 6.46·11-s + 3.95·13-s + 1.43·15-s − 3.43·17-s − 3.08·19-s + 4.41·21-s + 23-s + 25-s + 5.65·27-s + 0.863·29-s + 5.95·31-s − 9.26·33-s + 3.08·35-s − 7.03·37-s − 5.65·39-s + 5.60·41-s − 8·43-s + 0.950·45-s − 3.90·47-s + 2.53·49-s + 4.91·51-s − 6·53-s − 6.46·55-s + ⋯
L(s)  = 1  − 0.826·3-s − 0.447·5-s − 1.16·7-s − 0.316·9-s + 1.95·11-s + 1.09·13-s + 0.369·15-s − 0.832·17-s − 0.708·19-s + 0.964·21-s + 0.208·23-s + 0.200·25-s + 1.08·27-s + 0.160·29-s + 1.06·31-s − 1.61·33-s + 0.521·35-s − 1.15·37-s − 0.905·39-s + 0.875·41-s − 1.21·43-s + 0.141·45-s − 0.569·47-s + 0.361·49-s + 0.687·51-s − 0.824·53-s − 0.872·55-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 \( 1 + 1.43T + 3T^{2} \)
7 \( 1 + 3.08T + 7T^{2} \)
11 \( 1 - 6.46T + 11T^{2} \)
13 \( 1 - 3.95T + 13T^{2} \)
17 \( 1 + 3.43T + 17T^{2} \)
19 \( 1 + 3.08T + 19T^{2} \)
29 \( 1 - 0.863T + 29T^{2} \)
31 \( 1 - 5.95T + 31T^{2} \)
37 \( 1 + 7.03T + 37T^{2} \)
41 \( 1 - 5.60T + 41T^{2} \)
43 \( 1 + 8T + 43T^{2} \)
47 \( 1 + 3.90T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + 6.86T + 59T^{2} \)
61 \( 1 + 13.5T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 2.56T + 71T^{2} \)
73 \( 1 - 5.90T + 73T^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + 9.03T + 83T^{2} \)
89 \( 1 - 16.7T + 89T^{2} \)
97 \( 1 + 14.2T + 97T^{2} \)
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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

−8.903132206914549679217774284651, −8.267085106070241352926660708959, −6.72020924763452087665181835592, −6.57360718025229415866192774644, −5.94356142229100703688422690129, −4.65560005186299229420278065265, −3.85681347769988548252020946868, −3.04713768463449540823716191034, −1.35246250508633082812693008925, 0, 1.35246250508633082812693008925, 3.04713768463449540823716191034, 3.85681347769988548252020946868, 4.65560005186299229420278065265, 5.94356142229100703688422690129, 6.57360718025229415866192774644, 6.72020924763452087665181835592, 8.267085106070241352926660708959, 8.903132206914549679217774284651

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