Properties

Label 2-1840-1.1-c1-0-17
Degree $2$
Conductor $1840$
Sign $-1$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 5-s − 2·7-s + 6·9-s − 3·13-s + 3·15-s + 4·17-s + 4·19-s + 6·21-s + 23-s + 25-s − 9·27-s + 29-s − 31-s + 2·35-s − 8·37-s + 9·39-s + 11·41-s + 10·43-s − 6·45-s + 47-s − 3·49-s − 12·51-s − 6·53-s − 12·57-s + 8·59-s − 8·61-s + ⋯
L(s)  = 1  − 1.73·3-s − 0.447·5-s − 0.755·7-s + 2·9-s − 0.832·13-s + 0.774·15-s + 0.970·17-s + 0.917·19-s + 1.30·21-s + 0.208·23-s + 1/5·25-s − 1.73·27-s + 0.185·29-s − 0.179·31-s + 0.338·35-s − 1.31·37-s + 1.44·39-s + 1.71·41-s + 1.52·43-s − 0.894·45-s + 0.145·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.58·57-s + 1.04·59-s − 1.02·61-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: yes
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 \( 1 + p T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 - 11 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−9.109286203334659993096471269418, −7.65238981829781693182378402300, −7.24701157354504628844391503349, −6.33572440871088550887425929738, −5.63300398523041490404088005695, −4.96609072709435351263065669312, −4.03855439514098506279776346797, −2.92605729119644587705793314164, −1.15726237487066516293459663496, 0, 1.15726237487066516293459663496, 2.92605729119644587705793314164, 4.03855439514098506279776346797, 4.96609072709435351263065669312, 5.63300398523041490404088005695, 6.33572440871088550887425929738, 7.24701157354504628844391503349, 7.65238981829781693182378402300, 9.109286203334659993096471269418

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