Properties

Label 2-1815-1.1-c1-0-17
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  − 2.16·2-s + 3-s + 2.68·4-s + 5-s − 2.16·6-s − 0.480·7-s − 1.48·8-s + 9-s − 2.16·10-s + 2.68·12-s − 3.36·13-s + 1.03·14-s + 15-s − 2.16·16-s − 4.84·17-s − 2.16·18-s + 1.84·19-s + 2.68·20-s − 0.480·21-s + 3.48·23-s − 1.48·24-s + 25-s + 7.28·26-s + 27-s − 1.28·28-s + 8.32·29-s − 2.16·30-s + ⋯
L(s)  = 1  − 1.53·2-s + 0.577·3-s + 1.34·4-s + 0.447·5-s − 0.883·6-s − 0.181·7-s − 0.523·8-s + 0.333·9-s − 0.684·10-s + 0.774·12-s − 0.934·13-s + 0.277·14-s + 0.258·15-s − 0.541·16-s − 1.17·17-s − 0.510·18-s + 0.424·19-s + 0.600·20-s − 0.104·21-s + 0.725·23-s − 0.302·24-s + 0.200·25-s + 1.42·26-s + 0.192·27-s − 0.243·28-s + 1.54·29-s − 0.395·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9793963044\)
\(L(\frac12)\) \(\approx\) \(0.9793963044\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.16T + 2T^{2} \)
7 \( 1 + 0.480T + 7T^{2} \)
13 \( 1 + 3.36T + 13T^{2} \)
17 \( 1 + 4.84T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 3.48T + 23T^{2} \)
29 \( 1 - 8.32T + 29T^{2} \)
31 \( 1 + 2.36T + 31T^{2} \)
37 \( 1 - 5.44T + 37T^{2} \)
41 \( 1 - 6.65T + 41T^{2} \)
43 \( 1 + 4.32T + 43T^{2} \)
47 \( 1 + 3.17T + 47T^{2} \)
53 \( 1 - 13.5T + 53T^{2} \)
59 \( 1 - 15.0T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 - 8.17T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 5.84T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 5.67T + 89T^{2} \)
97 \( 1 + 8.17T + 97T^{2} \)
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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

−9.275465343018423949041030445203, −8.600454977200911188984058013287, −7.958971128028535731626569660208, −7.00883826678927327401722549710, −6.63704595542074971390252223261, −5.24225845762575283357219830946, −4.26611802124737412364292139433, −2.78516966713804589739700595851, −2.13136431246681253331907338546, −0.824133369501425312176330807248, 0.824133369501425312176330807248, 2.13136431246681253331907338546, 2.78516966713804589739700595851, 4.26611802124737412364292139433, 5.24225845762575283357219830946, 6.63704595542074971390252223261, 7.00883826678927327401722549710, 7.958971128028535731626569660208, 8.600454977200911188984058013287, 9.275465343018423949041030445203

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