Properties

Label 2-1856-1.1-c3-0-99
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.64·3-s − 12.8·5-s + 26.0·7-s − 5.41·9-s + 62.8·11-s − 22.3·13-s + 59.8·15-s − 57.9·17-s − 71.3·19-s − 121.·21-s − 49.5·23-s + 40.7·25-s + 150.·27-s + 29·29-s + 62.9·31-s − 291.·33-s − 335.·35-s − 119.·37-s + 104.·39-s − 414.·41-s + 348.·43-s + 69.7·45-s + 553.·47-s + 335.·49-s + 269.·51-s + 107.·53-s − 808.·55-s + ⋯
L(s)  = 1  − 0.894·3-s − 1.15·5-s + 1.40·7-s − 0.200·9-s + 1.72·11-s − 0.477·13-s + 1.02·15-s − 0.827·17-s − 0.861·19-s − 1.25·21-s − 0.449·23-s + 0.325·25-s + 1.07·27-s + 0.185·29-s + 0.364·31-s − 1.53·33-s − 1.61·35-s − 0.529·37-s + 0.427·39-s − 1.58·41-s + 1.23·43-s + 0.231·45-s + 1.71·47-s + 0.979·49-s + 0.739·51-s + 0.278·53-s − 1.98·55-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: \(1\)
Selberg data: \((2,\ 1856,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 4.64T + 27T^{2} \)
5 \( 1 + 12.8T + 125T^{2} \)
7 \( 1 - 26.0T + 343T^{2} \)
11 \( 1 - 62.8T + 1.33e3T^{2} \)
13 \( 1 + 22.3T + 2.19e3T^{2} \)
17 \( 1 + 57.9T + 4.91e3T^{2} \)
19 \( 1 + 71.3T + 6.85e3T^{2} \)
23 \( 1 + 49.5T + 1.21e4T^{2} \)
31 \( 1 - 62.9T + 2.97e4T^{2} \)
37 \( 1 + 119.T + 5.06e4T^{2} \)
41 \( 1 + 414.T + 6.89e4T^{2} \)
43 \( 1 - 348.T + 7.95e4T^{2} \)
47 \( 1 - 553.T + 1.03e5T^{2} \)
53 \( 1 - 107.T + 1.48e5T^{2} \)
59 \( 1 + 136.T + 2.05e5T^{2} \)
61 \( 1 - 579.T + 2.26e5T^{2} \)
67 \( 1 + 919.T + 3.00e5T^{2} \)
71 \( 1 - 781.T + 3.57e5T^{2} \)
73 \( 1 + 133.T + 3.89e5T^{2} \)
79 \( 1 - 868.T + 4.93e5T^{2} \)
83 \( 1 - 83.3T + 5.71e5T^{2} \)
89 \( 1 + 357.T + 7.04e5T^{2} \)
97 \( 1 + 187.T + 9.12e5T^{2} \)
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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

−8.549531680709104050566534409993, −7.68930020732539744120827903688, −6.86184905024920614343300702064, −6.15478506755053183201375239229, −5.08996763030716258396747875615, −4.38095073200934793466793782573, −3.81267031600309620110055640095, −2.22384003129698242774997197889, −1.07503721291844775694128717517, 0, 1.07503721291844775694128717517, 2.22384003129698242774997197889, 3.81267031600309620110055640095, 4.38095073200934793466793782573, 5.08996763030716258396747875615, 6.15478506755053183201375239229, 6.86184905024920614343300702064, 7.68930020732539744120827903688, 8.549531680709104050566534409993

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