Properties

Label 2-1859-1.1-c1-0-125
Degree $2$
Conductor $1859$
Sign $-1$
Analytic cond. $14.8441$
Root an. cond. $3.85281$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.92·2-s + 2.13·3-s + 1.70·4-s − 3.67·5-s + 4.10·6-s − 3.13·7-s − 0.571·8-s + 1.54·9-s − 7.07·10-s + 11-s + 3.63·12-s − 6.02·14-s − 7.84·15-s − 4.50·16-s + 3.22·17-s + 2.97·18-s − 3.26·19-s − 6.26·20-s − 6.67·21-s + 1.92·22-s − 8.55·23-s − 1.21·24-s + 8.52·25-s − 3.10·27-s − 5.33·28-s + 0.278·29-s − 15.0·30-s + ⋯
L(s)  = 1  + 1.36·2-s + 1.23·3-s + 0.851·4-s − 1.64·5-s + 1.67·6-s − 1.18·7-s − 0.201·8-s + 0.515·9-s − 2.23·10-s + 0.301·11-s + 1.04·12-s − 1.61·14-s − 2.02·15-s − 1.12·16-s + 0.783·17-s + 0.701·18-s − 0.749·19-s − 1.40·20-s − 1.45·21-s + 0.410·22-s − 1.78·23-s − 0.248·24-s + 1.70·25-s − 0.596·27-s − 1.00·28-s + 0.0516·29-s − 2.75·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1859\)    =    \(11 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(14.8441\)
Root analytic conductor: \(3.85281\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 - T \)
13 \( 1 \)
good2 \( 1 - 1.92T + 2T^{2} \)
3 \( 1 - 2.13T + 3T^{2} \)
5 \( 1 + 3.67T + 5T^{2} \)
7 \( 1 + 3.13T + 7T^{2} \)
17 \( 1 - 3.22T + 17T^{2} \)
19 \( 1 + 3.26T + 19T^{2} \)
23 \( 1 + 8.55T + 23T^{2} \)
29 \( 1 - 0.278T + 29T^{2} \)
31 \( 1 - 2.27T + 31T^{2} \)
37 \( 1 + 6.62T + 37T^{2} \)
41 \( 1 + 6.36T + 41T^{2} \)
43 \( 1 - 0.137T + 43T^{2} \)
47 \( 1 + 4.85T + 47T^{2} \)
53 \( 1 - 10.6T + 53T^{2} \)
59 \( 1 - 10.6T + 59T^{2} \)
61 \( 1 - 2.77T + 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 - 5.88T + 71T^{2} \)
73 \( 1 + 1.04T + 73T^{2} \)
79 \( 1 + 8.54T + 79T^{2} \)
83 \( 1 + 0.0460T + 83T^{2} \)
89 \( 1 + 1.86T + 89T^{2} \)
97 \( 1 + 3.72T + 97T^{2} \)
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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

−8.533449760660486678523767288240, −8.196654310608500666526021484255, −7.15216183520786865174854075803, −6.49961858702078854421027148980, −5.46678099670926821849446660450, −4.24799476341591778290082718656, −3.68776561001437932814641571554, −3.35131443325711520681954964269, −2.35577851091326134290065297043, 0, 2.35577851091326134290065297043, 3.35131443325711520681954964269, 3.68776561001437932814641571554, 4.24799476341591778290082718656, 5.46678099670926821849446660450, 6.49961858702078854421027148980, 7.15216183520786865174854075803, 8.196654310608500666526021484255, 8.533449760660486678523767288240

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