Properties

Label 2-1815-1.1-c1-0-38
Degree $2$
Conductor $1815$
Sign $-1$
Analytic cond. $14.4928$
Root an. cond. $3.80694$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2·4-s − 5-s − 1.73·7-s + 9-s + 2·12-s + 15-s + 4·16-s + 3.46·17-s + 5.19·19-s + 2·20-s + 1.73·21-s + 6·23-s + 25-s − 27-s + 3.46·28-s − 6.92·29-s + 31-s + 1.73·35-s − 2·36-s − 5·37-s − 3.46·41-s − 10.3·43-s − 45-s − 12·47-s − 4·48-s − 4·49-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s − 0.447·5-s − 0.654·7-s + 0.333·9-s + 0.577·12-s + 0.258·15-s + 16-s + 0.840·17-s + 1.19·19-s + 0.447·20-s + 0.377·21-s + 1.25·23-s + 0.200·25-s − 0.192·27-s + 0.654·28-s − 1.28·29-s + 0.179·31-s + 0.292·35-s − 0.333·36-s − 0.821·37-s − 0.541·41-s − 1.58·43-s − 0.149·45-s − 1.75·47-s − 0.577·48-s − 0.571·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $-1$
Analytic conductor: \(14.4928\)
Root analytic conductor: \(3.80694\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
5 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + 2T^{2} \)
7 \( 1 + 1.73T + 7T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 - 5.19T + 19T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 6.92T + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
37 \( 1 + 5T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 12T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 12.1T + 61T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 - 1.73T + 73T^{2} \)
79 \( 1 - 15.5T + 79T^{2} \)
83 \( 1 - 6.92T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 13T + 97T^{2} \)
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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

−8.967386097377301713078020815996, −8.096375974871344841906660400344, −7.29921933322431972523395961305, −6.48616862248567558759622718411, −5.30679273967680233329666995509, −5.04796426311487769793515776801, −3.72760061035960114855753063621, −3.23018612043223906164273475939, −1.25475203166947893171343135556, 0, 1.25475203166947893171343135556, 3.23018612043223906164273475939, 3.72760061035960114855753063621, 5.04796426311487769793515776801, 5.30679273967680233329666995509, 6.48616862248567558759622718411, 7.29921933322431972523395961305, 8.096375974871344841906660400344, 8.967386097377301713078020815996

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