Properties

Label 2-1815-1.1-c3-0-129
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  + 1.45·2-s − 3·3-s − 5.88·4-s + 5·5-s − 4.35·6-s − 20.0·7-s − 20.1·8-s + 9·9-s + 7.26·10-s + 17.6·12-s − 13.0·13-s − 29.1·14-s − 15·15-s + 17.7·16-s + 38.6·17-s + 13.0·18-s + 23.6·19-s − 29.4·20-s + 60.1·21-s + 63.5·23-s + 60.5·24-s + 25·25-s − 18.9·26-s − 27·27-s + 117.·28-s + 54.4·29-s − 21.7·30-s + ⋯
L(s)  = 1  + 0.513·2-s − 0.577·3-s − 0.736·4-s + 0.447·5-s − 0.296·6-s − 1.08·7-s − 0.891·8-s + 0.333·9-s + 0.229·10-s + 0.425·12-s − 0.278·13-s − 0.555·14-s − 0.258·15-s + 0.278·16-s + 0.551·17-s + 0.171·18-s + 0.285·19-s − 0.329·20-s + 0.624·21-s + 0.575·23-s + 0.514·24-s + 0.200·25-s − 0.143·26-s − 0.192·27-s + 0.796·28-s + 0.348·29-s − 0.132·30-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 \( 1 - 1.45T + 8T^{2} \)
7 \( 1 + 20.0T + 343T^{2} \)
13 \( 1 + 13.0T + 2.19e3T^{2} \)
17 \( 1 - 38.6T + 4.91e3T^{2} \)
19 \( 1 - 23.6T + 6.85e3T^{2} \)
23 \( 1 - 63.5T + 1.21e4T^{2} \)
29 \( 1 - 54.4T + 2.43e4T^{2} \)
31 \( 1 + 1.71T + 2.97e4T^{2} \)
37 \( 1 + 181.T + 5.06e4T^{2} \)
41 \( 1 - 88.1T + 6.89e4T^{2} \)
43 \( 1 - 414.T + 7.95e4T^{2} \)
47 \( 1 - 126.T + 1.03e5T^{2} \)
53 \( 1 + 161.T + 1.48e5T^{2} \)
59 \( 1 - 259.T + 2.05e5T^{2} \)
61 \( 1 - 427.T + 2.26e5T^{2} \)
67 \( 1 + 68.9T + 3.00e5T^{2} \)
71 \( 1 + 970.T + 3.57e5T^{2} \)
73 \( 1 + 601.T + 3.89e5T^{2} \)
79 \( 1 + 601.T + 4.93e5T^{2} \)
83 \( 1 - 197.T + 5.71e5T^{2} \)
89 \( 1 - 977.T + 7.04e5T^{2} \)
97 \( 1 - 1.04e3T + 9.12e5T^{2} \)
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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

−8.747822886148616718116338388436, −7.56751309939151830155258173984, −6.68810954253631947132212246834, −5.90797456081360760548891344755, −5.35042949705429950482030785883, −4.46183722641245440739538334002, −3.52704461853168230977992780305, −2.68738297992382123884367311562, −1.05491151753627855919594335701, 0, 1.05491151753627855919594335701, 2.68738297992382123884367311562, 3.52704461853168230977992780305, 4.46183722641245440739538334002, 5.35042949705429950482030785883, 5.90797456081360760548891344755, 6.68810954253631947132212246834, 7.56751309939151830155258173984, 8.747822886148616718116338388436

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