Properties

Label 2-1815-1.1-c3-0-114
Degree $2$
Conductor $1815$
Sign $1$
Analytic cond. $107.088$
Root an. cond. $10.3483$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5.06·2-s + 3·3-s + 17.6·4-s − 5·5-s + 15.1·6-s − 27.4·7-s + 48.8·8-s + 9·9-s − 25.3·10-s + 52.9·12-s − 22.6·13-s − 138.·14-s − 15·15-s + 106.·16-s + 41.1·17-s + 45.5·18-s + 142.·19-s − 88.2·20-s − 82.3·21-s + 176.·23-s + 146.·24-s + 25·25-s − 114.·26-s + 27·27-s − 484.·28-s − 76.2·29-s − 75.9·30-s + ⋯
L(s)  = 1  + 1.79·2-s + 0.577·3-s + 2.20·4-s − 0.447·5-s + 1.03·6-s − 1.48·7-s + 2.16·8-s + 0.333·9-s − 0.800·10-s + 1.27·12-s − 0.484·13-s − 2.65·14-s − 0.258·15-s + 1.66·16-s + 0.587·17-s + 0.596·18-s + 1.71·19-s − 0.986·20-s − 0.855·21-s + 1.59·23-s + 1.24·24-s + 0.200·25-s − 0.867·26-s + 0.192·27-s − 3.26·28-s − 0.487·29-s − 0.462·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1815 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1815\)    =    \(3 \cdot 5 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(107.088\)
Root analytic conductor: \(10.3483\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1815} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1815,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(7.573908424\)
\(L(\frac12)\) \(\approx\) \(7.573908424\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
11 \( 1 \)
good2 \( 1 - 5.06T + 8T^{2} \)
7 \( 1 + 27.4T + 343T^{2} \)
13 \( 1 + 22.6T + 2.19e3T^{2} \)
17 \( 1 - 41.1T + 4.91e3T^{2} \)
19 \( 1 - 142.T + 6.85e3T^{2} \)
23 \( 1 - 176.T + 1.21e4T^{2} \)
29 \( 1 + 76.2T + 2.43e4T^{2} \)
31 \( 1 - 197.T + 2.97e4T^{2} \)
37 \( 1 - 367.T + 5.06e4T^{2} \)
41 \( 1 - 238.T + 6.89e4T^{2} \)
43 \( 1 + 30.2T + 7.95e4T^{2} \)
47 \( 1 - 137.T + 1.03e5T^{2} \)
53 \( 1 - 638.T + 1.48e5T^{2} \)
59 \( 1 - 103.T + 2.05e5T^{2} \)
61 \( 1 + 605.T + 2.26e5T^{2} \)
67 \( 1 + 704.T + 3.00e5T^{2} \)
71 \( 1 + 782.T + 3.57e5T^{2} \)
73 \( 1 - 243.T + 3.89e5T^{2} \)
79 \( 1 - 532.T + 4.93e5T^{2} \)
83 \( 1 + 1.20e3T + 5.71e5T^{2} \)
89 \( 1 - 1.05e3T + 7.04e5T^{2} \)
97 \( 1 + 85.1T + 9.12e5T^{2} \)
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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

−9.037236598205921855574468569671, −7.56538731061693665212905206945, −7.26307822358259254531316984862, −6.33738140739165496773939797422, −5.58428701515979438719512434920, −4.68295404203727403511285468873, −3.81745474533621915538725886623, −2.99298786691303386945408748121, −2.75192513907620275509136818853, −0.977004267892786146990232190982, 0.977004267892786146990232190982, 2.75192513907620275509136818853, 2.99298786691303386945408748121, 3.81745474533621915538725886623, 4.68295404203727403511285468873, 5.58428701515979438719512434920, 6.33738140739165496773939797422, 7.26307822358259254531316984862, 7.56538731061693665212905206945, 9.037236598205921855574468569671

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