Properties

Label 2-1856-1.1-c1-0-42
Degree $2$
Conductor $1856$
Sign $-1$
Analytic cond. $14.8202$
Root an. cond. $3.84970$
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-s + 3·5-s − 2·7-s − 2·9-s − 3·11-s + 5·13-s − 3·15-s − 4·17-s + 2·21-s + 4·25-s + 5·27-s + 29-s − 9·31-s + 3·33-s − 6·35-s − 8·37-s − 5·39-s − 2·41-s − 11·43-s − 6·45-s + 7·47-s − 3·49-s + 4·51-s − 9·53-s − 9·55-s + 4·59-s + 12·61-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.34·5-s − 0.755·7-s − 2/3·9-s − 0.904·11-s + 1.38·13-s − 0.774·15-s − 0.970·17-s + 0.436·21-s + 4/5·25-s + 0.962·27-s + 0.185·29-s − 1.61·31-s + 0.522·33-s − 1.01·35-s − 1.31·37-s − 0.800·39-s − 0.312·41-s − 1.67·43-s − 0.894·45-s + 1.02·47-s − 3/7·49-s + 0.560·51-s − 1.23·53-s − 1.21·55-s + 0.520·59-s + 1.53·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $-1$
Analytic conductor: \(14.8202\)
Root analytic conductor: \(3.84970\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1856,\ (\ :1/2),\ -1)\)

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 \( 1 \)
29 \( 1 - T \)
good3 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 + 11 T + p T^{2} \)
47 \( 1 - 7 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−8.868470335391324440586597620218, −8.317920725627485157632811027240, −6.88127849884191867875357055362, −6.39208865343620833912323561954, −5.59484198644236369333229457444, −5.18505606705021347413228227439, −3.71393366615299243699613634856, −2.73069587768107365355477934045, −1.68175753467464429238376356886, 0, 1.68175753467464429238376356886, 2.73069587768107365355477934045, 3.71393366615299243699613634856, 5.18505606705021347413228227439, 5.59484198644236369333229457444, 6.39208865343620833912323561954, 6.88127849884191867875357055362, 8.317920725627485157632811027240, 8.868470335391324440586597620218

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