Properties

Label 2-1856-1.1-c1-0-36
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $14.8202$
Root an. cond. $3.84970$
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  + 2.76·3-s + 1.62·5-s + 4.62·9-s + 4.49·11-s − 0.103·13-s + 4.49·15-s + 2·17-s − 7.25·19-s + 5.52·23-s − 2.35·25-s + 4.49·27-s − 29-s + 6.76·31-s + 12.4·33-s − 5.25·37-s − 0.284·39-s + 5.79·41-s − 10.0·43-s + 7.52·45-s − 11.5·47-s − 7·49-s + 5.52·51-s + 7.14·53-s + 7.30·55-s − 20.0·57-s − 1.52·59-s − 9.04·61-s + ⋯
L(s)  = 1  + 1.59·3-s + 0.727·5-s + 1.54·9-s + 1.35·11-s − 0.0285·13-s + 1.15·15-s + 0.485·17-s − 1.66·19-s + 1.15·23-s − 0.471·25-s + 0.864·27-s − 0.185·29-s + 1.21·31-s + 2.15·33-s − 0.863·37-s − 0.0455·39-s + 0.904·41-s − 1.52·43-s + 1.12·45-s − 1.68·47-s − 49-s + 0.773·51-s + 0.982·53-s + 0.984·55-s − 2.65·57-s − 0.198·59-s − 1.15·61-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: no
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.735031818\)
\(L(\frac12)\) \(\approx\) \(3.735031818\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.76T + 3T^{2} \)
5 \( 1 - 1.62T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 4.49T + 11T^{2} \)
13 \( 1 + 0.103T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 7.25T + 19T^{2} \)
23 \( 1 - 5.52T + 23T^{2} \)
31 \( 1 - 6.76T + 31T^{2} \)
37 \( 1 + 5.25T + 37T^{2} \)
41 \( 1 - 5.79T + 41T^{2} \)
43 \( 1 + 10.0T + 43T^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 - 7.14T + 53T^{2} \)
59 \( 1 + 1.52T + 59T^{2} \)
61 \( 1 + 9.04T + 61T^{2} \)
67 \( 1 - 15.0T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 + 1.79T + 73T^{2} \)
79 \( 1 - 1.98T + 79T^{2} \)
83 \( 1 - 6.47T + 83T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 - 1.25T + 97T^{2} \)
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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

−9.262744918780497469711619451803, −8.489016833877073401258824861680, −7.981171681544970763320936435134, −6.77974615302055051431553914110, −6.37727271131036067042078777118, −5.02408656606041535181831260705, −4.02018853224252054978530500416, −3.28094142713394836570320414246, −2.25549755509276349133947459075, −1.45923667950737054077261786499, 1.45923667950737054077261786499, 2.25549755509276349133947459075, 3.28094142713394836570320414246, 4.02018853224252054978530500416, 5.02408656606041535181831260705, 6.37727271131036067042078777118, 6.77974615302055051431553914110, 7.981171681544970763320936435134, 8.489016833877073401258824861680, 9.262744918780497469711619451803

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