Properties

Label 2-1840-460.459-c1-0-14
Degree $2$
Conductor $1840$
Sign $-0.147 - 0.989i$
Analytic cond. $14.6924$
Root an. cond. $3.83307$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.16·3-s − 2.23i·5-s + 4.24i·7-s + 7.00·9-s + 7.07i·15-s − 13.4i·21-s + (4.74 − 0.707i)23-s − 5.00·25-s − 12.6·27-s − 6·29-s + 9.48·35-s + 12·41-s − 12.7i·43-s − 15.6i·45-s − 9.48·47-s + ⋯
L(s)  = 1  − 1.82·3-s − 0.999i·5-s + 1.60i·7-s + 2.33·9-s + 1.82i·15-s − 2.92i·21-s + (0.989 − 0.147i)23-s − 1.00·25-s − 2.43·27-s − 1.11·29-s + 1.60·35-s + 1.87·41-s − 1.94i·43-s − 2.33i·45-s − 1.38·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.147 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1840\)    =    \(2^{4} \cdot 5 \cdot 23\)
Sign: $-0.147 - 0.989i$
Analytic conductor: \(14.6924\)
Root analytic conductor: \(3.83307\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1840} (1839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1840,\ (\ :1/2),\ -0.147 - 0.989i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5374808068\)
\(L(\frac12)\) \(\approx\) \(0.5374808068\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.23iT \)
23 \( 1 + (-4.74 + 0.707i)T \)
good3 \( 1 + 3.16T + 3T^{2} \)
7 \( 1 - 4.24iT - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 19T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 + 12.7iT - 43T^{2} \)
47 \( 1 + 9.48T + 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 - 4.24iT - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 17.8iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−9.352822479559527115912665098618, −8.934018647053878880677392408365, −7.84943738465402844405477274870, −6.84551650554756518116191131382, −5.95054161269808339431890328114, −5.44125879953846344446454867939, −4.98576917281967103727691240206, −3.97095303718592121737034638264, −2.26038205265691797041900241180, −1.05077269191663283650414639431, 0.31706902330945866442491259851, 1.50523716604757432411317030971, 3.29590481158386921081225895181, 4.24622284453921309041975398225, 4.94654517481940278330848550806, 6.05332309495927469186098916214, 6.53030672977214159267916621211, 7.34741862808484846566094614440, 7.68286440594562655718242356795, 9.475031646340379898794273593460

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