Properties

Label 2-1856-1.1-c1-0-5
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  − 2.12·3-s − 1.48·5-s + 1.51·9-s + 3.15·11-s − 6.76·13-s + 3.15·15-s + 2·17-s − 1.03·19-s − 4.24·23-s − 2.79·25-s + 3.15·27-s − 29-s + 1.87·31-s − 6.70·33-s + 0.969·37-s + 14.3·39-s − 7.52·41-s + 1.09·43-s − 2.24·45-s + 9.34·47-s − 7·49-s − 4.24·51-s − 5.73·53-s − 4.68·55-s + 2.18·57-s + 8.24·59-s + 10.4·61-s + ⋯
L(s)  = 1  − 1.22·3-s − 0.664·5-s + 0.505·9-s + 0.951·11-s − 1.87·13-s + 0.814·15-s + 0.485·17-s − 0.236·19-s − 0.886·23-s − 0.559·25-s + 0.607·27-s − 0.185·29-s + 0.336·31-s − 1.16·33-s + 0.159·37-s + 2.30·39-s − 1.17·41-s + 0.166·43-s − 0.335·45-s + 1.36·47-s − 49-s − 0.595·51-s − 0.787·53-s − 0.631·55-s + 0.289·57-s + 1.07·59-s + 1.34·61-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

\(L(1)\) \(\approx\) \(0.6219894350\)
\(L(\frac12)\) \(\approx\) \(0.6219894350\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.12T + 3T^{2} \)
5 \( 1 + 1.48T + 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 3.15T + 11T^{2} \)
13 \( 1 + 6.76T + 13T^{2} \)
17 \( 1 - 2T + 17T^{2} \)
19 \( 1 + 1.03T + 19T^{2} \)
23 \( 1 + 4.24T + 23T^{2} \)
31 \( 1 - 1.87T + 31T^{2} \)
37 \( 1 - 0.969T + 37T^{2} \)
41 \( 1 + 7.52T + 41T^{2} \)
43 \( 1 - 1.09T + 43T^{2} \)
47 \( 1 - 9.34T + 47T^{2} \)
53 \( 1 + 5.73T + 53T^{2} \)
59 \( 1 - 8.24T + 59T^{2} \)
61 \( 1 - 10.4T + 61T^{2} \)
67 \( 1 + 4.49T + 67T^{2} \)
71 \( 1 + 10.1T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 - 13.5T + 89T^{2} \)
97 \( 1 + 4.96T + 97T^{2} \)
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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

−9.465468168476728427348750021680, −8.324018514517283215030725231430, −7.54007372699774709849394215876, −6.80491798489944905710564185999, −6.04651957334448189316370596550, −5.16358704870627051564025232693, −4.47976621685674879321493778805, −3.52689088308962823922966992057, −2.12752472267340193791689713496, −0.54978540677904334919663387981, 0.54978540677904334919663387981, 2.12752472267340193791689713496, 3.52689088308962823922966992057, 4.47976621685674879321493778805, 5.16358704870627051564025232693, 6.04651957334448189316370596550, 6.80491798489944905710564185999, 7.54007372699774709849394215876, 8.324018514517283215030725231430, 9.465468168476728427348750021680

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