Properties

Label 2-1856-1.1-c3-0-167
Degree $2$
Conductor $1856$
Sign $1$
Analytic cond. $109.507$
Root an. cond. $10.4645$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $2$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 7·3-s + 13·5-s − 16·7-s + 22·9-s − 45·11-s − 61·13-s − 91·15-s − 102·17-s − 68·19-s + 112·21-s − 194·23-s + 44·25-s + 35·27-s + 29·29-s − 149·31-s + 315·33-s − 208·35-s − 400·37-s + 427·39-s + 280·41-s + 263·43-s + 286·45-s − 509·47-s − 87·49-s + 714·51-s + 605·53-s − 585·55-s + ⋯
L(s)  = 1  − 1.34·3-s + 1.16·5-s − 0.863·7-s + 0.814·9-s − 1.23·11-s − 1.30·13-s − 1.56·15-s − 1.45·17-s − 0.821·19-s + 1.16·21-s − 1.75·23-s + 0.351·25-s + 0.249·27-s + 0.185·29-s − 0.863·31-s + 1.66·33-s − 1.00·35-s − 1.77·37-s + 1.75·39-s + 1.06·41-s + 0.932·43-s + 0.947·45-s − 1.57·47-s − 0.253·49-s + 1.96·51-s + 1.56·53-s − 1.43·55-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: yes
Arithmetic: yes
Character: $\chi_{1856} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(2\)
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 \( 1 - p T \)
good3 \( 1 + 7 T + p^{3} T^{2} \)
5 \( 1 - 13 T + p^{3} T^{2} \)
7 \( 1 + 16 T + p^{3} T^{2} \)
11 \( 1 + 45 T + p^{3} T^{2} \)
13 \( 1 + 61 T + p^{3} T^{2} \)
17 \( 1 + 6 p T + p^{3} T^{2} \)
19 \( 1 + 68 T + p^{3} T^{2} \)
23 \( 1 + 194 T + p^{3} T^{2} \)
31 \( 1 + 149 T + p^{3} T^{2} \)
37 \( 1 + 400 T + p^{3} T^{2} \)
41 \( 1 - 280 T + p^{3} T^{2} \)
43 \( 1 - 263 T + p^{3} T^{2} \)
47 \( 1 + 509 T + p^{3} T^{2} \)
53 \( 1 - 605 T + p^{3} T^{2} \)
59 \( 1 + 578 T + p^{3} T^{2} \)
61 \( 1 - 718 T + p^{3} T^{2} \)
67 \( 1 + 260 T + p^{3} T^{2} \)
71 \( 1 + 738 T + p^{3} T^{2} \)
73 \( 1 - 652 T + p^{3} T^{2} \)
79 \( 1 - 917 T + p^{3} T^{2} \)
83 \( 1 - 678 T + p^{3} T^{2} \)
89 \( 1 + 1008 T + p^{3} T^{2} \)
97 \( 1 + 1764 T + p^{3} T^{2} \)
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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

−8.025417810235534177782706988453, −6.91790443165846695146659537877, −6.40923224522338199398662700050, −5.64575707789603546714428899488, −5.14037201446945329063283619306, −4.16870872831896446982436758049, −2.59973859217988199322172260845, −1.97109772370186725936697464255, 0, 0, 1.97109772370186725936697464255, 2.59973859217988199322172260845, 4.16870872831896446982436758049, 5.14037201446945329063283619306, 5.64575707789603546714428899488, 6.40923224522338199398662700050, 6.91790443165846695146659537877, 8.025417810235534177782706988453

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