Properties

Label 2-1815-1.1-c3-0-137
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0881·2-s + 3·3-s − 7.99·4-s − 5·5-s − 0.264·6-s − 20.0·7-s + 1.41·8-s + 9·9-s + 0.440·10-s − 23.9·12-s + 19.8·13-s + 1.77·14-s − 15·15-s + 63.8·16-s + 7.50·17-s − 0.793·18-s + 27.5·19-s + 39.9·20-s − 60.2·21-s − 27.3·23-s + 4.23·24-s + 25·25-s − 1.74·26-s + 27·27-s + 160.·28-s − 98.3·29-s + 1.32·30-s + ⋯
L(s)  = 1  − 0.0311·2-s + 0.577·3-s − 0.999·4-s − 0.447·5-s − 0.0179·6-s − 1.08·7-s + 0.0623·8-s + 0.333·9-s + 0.0139·10-s − 0.576·12-s + 0.422·13-s + 0.0338·14-s − 0.258·15-s + 0.997·16-s + 0.107·17-s − 0.0103·18-s + 0.332·19-s + 0.446·20-s − 0.626·21-s − 0.248·23-s + 0.0359·24-s + 0.200·25-s − 0.0131·26-s + 0.192·27-s + 1.08·28-s − 0.629·29-s + 0.00804·30-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1815,\ (\ :3/2),\ -1)\)

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)$
bad3 \( 1 - 3T \)
5 \( 1 + 5T \)
11 \( 1 \)
good2 \( 1 + 0.0881T + 8T^{2} \)
7 \( 1 + 20.0T + 343T^{2} \)
13 \( 1 - 19.8T + 2.19e3T^{2} \)
17 \( 1 - 7.50T + 4.91e3T^{2} \)
19 \( 1 - 27.5T + 6.85e3T^{2} \)
23 \( 1 + 27.3T + 1.21e4T^{2} \)
29 \( 1 + 98.3T + 2.43e4T^{2} \)
31 \( 1 - 88.1T + 2.97e4T^{2} \)
37 \( 1 + 53.2T + 5.06e4T^{2} \)
41 \( 1 - 197.T + 6.89e4T^{2} \)
43 \( 1 + 37.1T + 7.95e4T^{2} \)
47 \( 1 - 528.T + 1.03e5T^{2} \)
53 \( 1 + 306.T + 1.48e5T^{2} \)
59 \( 1 + 69.0T + 2.05e5T^{2} \)
61 \( 1 - 779.T + 2.26e5T^{2} \)
67 \( 1 - 911.T + 3.00e5T^{2} \)
71 \( 1 + 100.T + 3.57e5T^{2} \)
73 \( 1 + 173.T + 3.89e5T^{2} \)
79 \( 1 + 183.T + 4.93e5T^{2} \)
83 \( 1 + 1.14e3T + 5.71e5T^{2} \)
89 \( 1 - 1.16e3T + 7.04e5T^{2} \)
97 \( 1 + 1.20e3T + 9.12e5T^{2} \)
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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

−8.576712562193838920500461407779, −7.891017011625802491851814626045, −7.05665970174207941360582931939, −6.09225921027473624425431942479, −5.19013428714768207409532449051, −4.07150746441515577809067854367, −3.61175897445654334245236361920, −2.64770600753337033924852657964, −1.08156549464926757125270584885, 0, 1.08156549464926757125270584885, 2.64770600753337033924852657964, 3.61175897445654334245236361920, 4.07150746441515577809067854367, 5.19013428714768207409532449051, 6.09225921027473624425431942479, 7.05665970174207941360582931939, 7.891017011625802491851814626045, 8.576712562193838920500461407779

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