Properties

Label 2-1856-1.1-c3-0-108
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  − 9.04·3-s − 2.15·5-s + 27.7·7-s + 54.7·9-s + 0.351·11-s + 65.3·13-s + 19.5·15-s − 68.3·17-s − 89.8·19-s − 251.·21-s + 110.·23-s − 120.·25-s − 251.·27-s + 29·29-s − 258.·31-s − 3.17·33-s − 59.9·35-s + 128.·37-s − 591.·39-s + 283.·41-s − 339.·43-s − 118.·45-s − 147.·47-s + 429.·49-s + 617.·51-s − 518.·53-s − 0.758·55-s + ⋯
L(s)  = 1  − 1.74·3-s − 0.192·5-s + 1.50·7-s + 2.02·9-s + 0.00963·11-s + 1.39·13-s + 0.335·15-s − 0.974·17-s − 1.08·19-s − 2.61·21-s + 1.00·23-s − 0.962·25-s − 1.79·27-s + 0.185·29-s − 1.49·31-s − 0.0167·33-s − 0.289·35-s + 0.569·37-s − 2.42·39-s + 1.07·41-s − 1.20·43-s − 0.391·45-s − 0.457·47-s + 1.25·49-s + 1.69·51-s − 1.34·53-s − 0.00185·55-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 \( 1 + 9.04T + 27T^{2} \)
5 \( 1 + 2.15T + 125T^{2} \)
7 \( 1 - 27.7T + 343T^{2} \)
11 \( 1 - 0.351T + 1.33e3T^{2} \)
13 \( 1 - 65.3T + 2.19e3T^{2} \)
17 \( 1 + 68.3T + 4.91e3T^{2} \)
19 \( 1 + 89.8T + 6.85e3T^{2} \)
23 \( 1 - 110.T + 1.21e4T^{2} \)
31 \( 1 + 258.T + 2.97e4T^{2} \)
37 \( 1 - 128.T + 5.06e4T^{2} \)
41 \( 1 - 283.T + 6.89e4T^{2} \)
43 \( 1 + 339.T + 7.95e4T^{2} \)
47 \( 1 + 147.T + 1.03e5T^{2} \)
53 \( 1 + 518.T + 1.48e5T^{2} \)
59 \( 1 - 102.T + 2.05e5T^{2} \)
61 \( 1 - 791.T + 2.26e5T^{2} \)
67 \( 1 + 287.T + 3.00e5T^{2} \)
71 \( 1 + 546.T + 3.57e5T^{2} \)
73 \( 1 + 260.T + 3.89e5T^{2} \)
79 \( 1 + 204.T + 4.93e5T^{2} \)
83 \( 1 - 949.T + 5.71e5T^{2} \)
89 \( 1 - 1.39e3T + 7.04e5T^{2} \)
97 \( 1 - 1.22e3T + 9.12e5T^{2} \)
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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

−8.424392531478428416496434244411, −7.62054559699169812468325615316, −6.66976815931012923440983400960, −6.06219818410317185581475606634, −5.23163533675232470460631608890, −4.57289847072233145822261584332, −3.85631844390744526384646516353, −1.95371764650103523748548983196, −1.15143570614474843517151662041, 0, 1.15143570614474843517151662041, 1.95371764650103523748548983196, 3.85631844390744526384646516353, 4.57289847072233145822261584332, 5.23163533675232470460631608890, 6.06219818410317185581475606634, 6.66976815931012923440983400960, 7.62054559699169812468325615316, 8.424392531478428416496434244411

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