Properties

Label 2-1859-1.1-c3-0-217
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $109.684$
Root an. cond. $10.4730$
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  + 2.06·2-s − 3.79·3-s − 3.72·4-s − 11.8·5-s − 7.84·6-s + 6.57·7-s − 24.2·8-s − 12.6·9-s − 24.4·10-s − 11·11-s + 14.1·12-s + 13.5·14-s + 44.7·15-s − 20.3·16-s + 37.1·17-s − 26.0·18-s + 85.9·19-s + 43.9·20-s − 24.9·21-s − 22.7·22-s + 55.7·23-s + 91.9·24-s + 14.5·25-s + 150.·27-s − 24.4·28-s + 67.2·29-s + 92.6·30-s + ⋯
L(s)  = 1  + 0.731·2-s − 0.729·3-s − 0.465·4-s − 1.05·5-s − 0.533·6-s + 0.354·7-s − 1.07·8-s − 0.467·9-s − 0.772·10-s − 0.301·11-s + 0.339·12-s + 0.259·14-s + 0.771·15-s − 0.318·16-s + 0.530·17-s − 0.341·18-s + 1.03·19-s + 0.491·20-s − 0.258·21-s − 0.220·22-s + 0.505·23-s + 0.782·24-s + 0.116·25-s + 1.07·27-s − 0.165·28-s + 0.430·29-s + 0.563·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(109.684\)
Root analytic conductor: \(10.4730\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :3/2),\ -1)\)

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)$
bad11 \( 1 + 11T \)
13 \( 1 \)
good2 \( 1 - 2.06T + 8T^{2} \)
3 \( 1 + 3.79T + 27T^{2} \)
5 \( 1 + 11.8T + 125T^{2} \)
7 \( 1 - 6.57T + 343T^{2} \)
17 \( 1 - 37.1T + 4.91e3T^{2} \)
19 \( 1 - 85.9T + 6.85e3T^{2} \)
23 \( 1 - 55.7T + 1.21e4T^{2} \)
29 \( 1 - 67.2T + 2.43e4T^{2} \)
31 \( 1 + 92.3T + 2.97e4T^{2} \)
37 \( 1 - 100.T + 5.06e4T^{2} \)
41 \( 1 - 97.9T + 6.89e4T^{2} \)
43 \( 1 + 79.2T + 7.95e4T^{2} \)
47 \( 1 - 204.T + 1.03e5T^{2} \)
53 \( 1 - 76.1T + 1.48e5T^{2} \)
59 \( 1 - 14.4T + 2.05e5T^{2} \)
61 \( 1 - 254.T + 2.26e5T^{2} \)
67 \( 1 + 220.T + 3.00e5T^{2} \)
71 \( 1 - 399.T + 3.57e5T^{2} \)
73 \( 1 - 664.T + 3.89e5T^{2} \)
79 \( 1 - 222.T + 4.93e5T^{2} \)
83 \( 1 + 1.24e3T + 5.71e5T^{2} \)
89 \( 1 + 1.21e3T + 7.04e5T^{2} \)
97 \( 1 + 569.T + 9.12e5T^{2} \)
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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

−8.350046902932946934547084302547, −7.75214933229829195210139172666, −6.77637398337740519916030347377, −5.72196281972556856851364211182, −5.23833603750788130186975100813, −4.47077817936415339033081207130, −3.59592312647207936102190319863, −2.81118205619891263421294769988, −0.939194156864233755626836636126, 0, 0.939194156864233755626836636126, 2.81118205619891263421294769988, 3.59592312647207936102190319863, 4.47077817936415339033081207130, 5.23833603750788130186975100813, 5.72196281972556856851364211182, 6.77637398337740519916030347377, 7.75214933229829195210139172666, 8.350046902932946934547084302547

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