Properties

Label 2-1815-1.1-c1-0-40
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.47·2-s + 3-s + 0.182·4-s + 5-s + 1.47·6-s + 2.29·7-s − 2.68·8-s + 9-s + 1.47·10-s + 0.182·12-s + 2.14·13-s + 3.39·14-s + 15-s − 4.33·16-s − 0.544·17-s + 1.47·18-s + 2.18·19-s + 0.182·20-s + 2.29·21-s + 2.03·23-s − 2.68·24-s + 25-s + 3.16·26-s + 27-s + 0.418·28-s + 9.94·29-s + 1.47·30-s + ⋯
L(s)  = 1  + 1.04·2-s + 0.577·3-s + 0.0911·4-s + 0.447·5-s + 0.603·6-s + 0.867·7-s − 0.949·8-s + 0.333·9-s + 0.467·10-s + 0.0526·12-s + 0.593·13-s + 0.906·14-s + 0.258·15-s − 1.08·16-s − 0.132·17-s + 0.348·18-s + 0.500·19-s + 0.0407·20-s + 0.500·21-s + 0.425·23-s − 0.548·24-s + 0.200·25-s + 0.620·26-s + 0.192·27-s + 0.0790·28-s + 1.84·29-s + 0.269·30-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: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.985539922\)
\(L(\frac12)\) \(\approx\) \(3.985539922\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.47T + 2T^{2} \)
7 \( 1 - 2.29T + 7T^{2} \)
13 \( 1 - 2.14T + 13T^{2} \)
17 \( 1 + 0.544T + 17T^{2} \)
19 \( 1 - 2.18T + 19T^{2} \)
23 \( 1 - 2.03T + 23T^{2} \)
29 \( 1 - 9.94T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 + 8.81T + 37T^{2} \)
41 \( 1 - 1.82T + 41T^{2} \)
43 \( 1 - 0.620T + 43T^{2} \)
47 \( 1 + 0.378T + 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 + 8.07T + 59T^{2} \)
61 \( 1 + 8.72T + 61T^{2} \)
67 \( 1 + 9.75T + 67T^{2} \)
71 \( 1 + 14.9T + 71T^{2} \)
73 \( 1 + 7.85T + 73T^{2} \)
79 \( 1 - 9.22T + 79T^{2} \)
83 \( 1 - 9.45T + 83T^{2} \)
89 \( 1 - 0.583T + 89T^{2} \)
97 \( 1 + 5.37T + 97T^{2} \)
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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

−8.996771396087182805863567950655, −8.646548466705989431249614376016, −7.72430851673907460203923360645, −6.66678283705470563205492839797, −5.92319534438457257042271176238, −4.94081810450433488104508055153, −4.47558276610164603896235502731, −3.37459815923837553501007013338, −2.59950914380238059770492480143, −1.29220227461572524215152317490, 1.29220227461572524215152317490, 2.59950914380238059770492480143, 3.37459815923837553501007013338, 4.47558276610164603896235502731, 4.94081810450433488104508055153, 5.92319534438457257042271176238, 6.66678283705470563205492839797, 7.72430851673907460203923360645, 8.646548466705989431249614376016, 8.996771396087182805863567950655

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