Properties

Label 2-1856-1.1-c3-0-118
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 9.24·3-s − 0.656·5-s + 6.14·7-s + 58.4·9-s + 65.3·11-s + 49.7·13-s − 6.07·15-s + 55.4·17-s + 64.7·19-s + 56.7·21-s + 93.8·23-s − 124.·25-s + 290.·27-s − 29·29-s − 236.·31-s + 603.·33-s − 4.03·35-s − 76.8·37-s + 460.·39-s + 215.·41-s − 80.8·43-s − 38.3·45-s − 357.·47-s − 305.·49-s + 512.·51-s − 328.·53-s − 42.9·55-s + ⋯
L(s)  = 1  + 1.77·3-s − 0.0587·5-s + 0.331·7-s + 2.16·9-s + 1.79·11-s + 1.06·13-s − 0.104·15-s + 0.791·17-s + 0.781·19-s + 0.589·21-s + 0.851·23-s − 0.996·25-s + 2.07·27-s − 0.185·29-s − 1.36·31-s + 3.18·33-s − 0.0194·35-s − 0.341·37-s + 1.88·39-s + 0.819·41-s − 0.286·43-s − 0.127·45-s − 1.11·47-s − 0.890·49-s + 1.40·51-s − 0.851·53-s − 0.105·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1856\)    =    \(2^{6} \cdot 29\)
Sign: $1$
Analytic conductor: \(109.507\)
Root analytic conductor: \(10.4645\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(6.082285017\)
\(L(\frac12)\) \(\approx\) \(6.082285017\)
\(L(\frac{5}{2})\) 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 \( 1 + 29T \)
good3 \( 1 - 9.24T + 27T^{2} \)
5 \( 1 + 0.656T + 125T^{2} \)
7 \( 1 - 6.14T + 343T^{2} \)
11 \( 1 - 65.3T + 1.33e3T^{2} \)
13 \( 1 - 49.7T + 2.19e3T^{2} \)
17 \( 1 - 55.4T + 4.91e3T^{2} \)
19 \( 1 - 64.7T + 6.85e3T^{2} \)
23 \( 1 - 93.8T + 1.21e4T^{2} \)
31 \( 1 + 236.T + 2.97e4T^{2} \)
37 \( 1 + 76.8T + 5.06e4T^{2} \)
41 \( 1 - 215.T + 6.89e4T^{2} \)
43 \( 1 + 80.8T + 7.95e4T^{2} \)
47 \( 1 + 357.T + 1.03e5T^{2} \)
53 \( 1 + 328.T + 1.48e5T^{2} \)
59 \( 1 - 99.2T + 2.05e5T^{2} \)
61 \( 1 - 725.T + 2.26e5T^{2} \)
67 \( 1 + 844.T + 3.00e5T^{2} \)
71 \( 1 + 378.T + 3.57e5T^{2} \)
73 \( 1 + 581.T + 3.89e5T^{2} \)
79 \( 1 + 353.T + 4.93e5T^{2} \)
83 \( 1 + 696.T + 5.71e5T^{2} \)
89 \( 1 - 1.11e3T + 7.04e5T^{2} \)
97 \( 1 + 805.T + 9.12e5T^{2} \)
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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

−8.955990410472242526800123499670, −8.198121888834651942471490852510, −7.49914669265196095635401410743, −6.77436844831257880317041922202, −5.72054936773169372076618004715, −4.40695474363111085089052042324, −3.61672943294744400938941266858, −3.19597087224640337507455113700, −1.75395916565761060623624003151, −1.25614658876055763001344382665, 1.25614658876055763001344382665, 1.75395916565761060623624003151, 3.19597087224640337507455113700, 3.61672943294744400938941266858, 4.40695474363111085089052042324, 5.72054936773169372076618004715, 6.77436844831257880317041922202, 7.49914669265196095635401410743, 8.198121888834651942471490852510, 8.955990410472242526800123499670

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