Properties

Label 2-1856-1.1-c3-0-102
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  + 3.81·3-s − 15.9·5-s − 24.5·7-s − 12.4·9-s + 12.0·11-s + 50.0·13-s − 60.6·15-s + 40.0·17-s + 138.·19-s − 93.5·21-s + 68.4·23-s + 128.·25-s − 150.·27-s − 29·29-s + 26.3·31-s + 45.8·33-s + 390.·35-s − 337.·37-s + 190.·39-s + 459.·41-s − 64.3·43-s + 198.·45-s − 492.·47-s + 260.·49-s + 152.·51-s + 319.·53-s − 191.·55-s + ⋯
L(s)  = 1  + 0.733·3-s − 1.42·5-s − 1.32·7-s − 0.462·9-s + 0.329·11-s + 1.06·13-s − 1.04·15-s + 0.571·17-s + 1.66·19-s − 0.972·21-s + 0.620·23-s + 1.02·25-s − 1.07·27-s − 0.185·29-s + 0.152·31-s + 0.241·33-s + 1.88·35-s − 1.49·37-s + 0.783·39-s + 1.74·41-s − 0.228·43-s + 0.657·45-s − 1.52·47-s + 0.758·49-s + 0.418·51-s + 0.828·53-s − 0.469·55-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: Trivial
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 \( 1 - 3.81T + 27T^{2} \)
5 \( 1 + 15.9T + 125T^{2} \)
7 \( 1 + 24.5T + 343T^{2} \)
11 \( 1 - 12.0T + 1.33e3T^{2} \)
13 \( 1 - 50.0T + 2.19e3T^{2} \)
17 \( 1 - 40.0T + 4.91e3T^{2} \)
19 \( 1 - 138.T + 6.85e3T^{2} \)
23 \( 1 - 68.4T + 1.21e4T^{2} \)
31 \( 1 - 26.3T + 2.97e4T^{2} \)
37 \( 1 + 337.T + 5.06e4T^{2} \)
41 \( 1 - 459.T + 6.89e4T^{2} \)
43 \( 1 + 64.3T + 7.95e4T^{2} \)
47 \( 1 + 492.T + 1.03e5T^{2} \)
53 \( 1 - 319.T + 1.48e5T^{2} \)
59 \( 1 + 284.T + 2.05e5T^{2} \)
61 \( 1 - 326.T + 2.26e5T^{2} \)
67 \( 1 + 326.T + 3.00e5T^{2} \)
71 \( 1 - 94.1T + 3.57e5T^{2} \)
73 \( 1 - 134.T + 3.89e5T^{2} \)
79 \( 1 - 641.T + 4.93e5T^{2} \)
83 \( 1 + 850.T + 5.71e5T^{2} \)
89 \( 1 + 93.6T + 7.04e5T^{2} \)
97 \( 1 - 740.T + 9.12e5T^{2} \)
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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

−8.491338528396072483441170017645, −7.75316607565657934637221458107, −7.10363826699049062242842742543, −6.19370437233463610551761122894, −5.24666871973811796470464411289, −3.85727873693589959545516378231, −3.45028749218948284338753882360, −2.87116460056790833000587311536, −1.11481032526215678960545477382, 0, 1.11481032526215678960545477382, 2.87116460056790833000587311536, 3.45028749218948284338753882360, 3.85727873693589959545516378231, 5.24666871973811796470464411289, 6.19370437233463610551761122894, 7.10363826699049062242842742543, 7.75316607565657934637221458107, 8.491338528396072483441170017645

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