Properties

Label 2-1859-1.1-c1-0-111
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·2-s + 1.36·3-s + 2.98·4-s + 4.28·5-s + 3.03·6-s + 0.384·7-s + 2.19·8-s − 1.14·9-s + 9.56·10-s + 11-s + 4.06·12-s + 0.857·14-s + 5.83·15-s − 1.07·16-s − 4.75·17-s − 2.55·18-s − 2.17·19-s + 12.7·20-s + 0.523·21-s + 2.23·22-s + 2.70·23-s + 2.98·24-s + 13.3·25-s − 5.64·27-s + 1.14·28-s − 10.2·29-s + 13.0·30-s + ⋯
L(s)  = 1  + 1.57·2-s + 0.786·3-s + 1.49·4-s + 1.91·5-s + 1.24·6-s + 0.145·7-s + 0.774·8-s − 0.381·9-s + 3.02·10-s + 0.301·11-s + 1.17·12-s + 0.229·14-s + 1.50·15-s − 0.268·16-s − 1.15·17-s − 0.602·18-s − 0.498·19-s + 2.85·20-s + 0.114·21-s + 0.475·22-s + 0.563·23-s + 0.609·24-s + 2.67·25-s − 1.08·27-s + 0.216·28-s − 1.90·29-s + 2.37·30-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: $\chi_{1859} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 1859,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.928316829\)
\(L(\frac12)\) \(\approx\) \(6.928316829\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T + 2T^{2} \)
3 \( 1 - 1.36T + 3T^{2} \)
5 \( 1 - 4.28T + 5T^{2} \)
7 \( 1 - 0.384T + 7T^{2} \)
17 \( 1 + 4.75T + 17T^{2} \)
19 \( 1 + 2.17T + 19T^{2} \)
23 \( 1 - 2.70T + 23T^{2} \)
29 \( 1 + 10.2T + 29T^{2} \)
31 \( 1 - 3.64T + 31T^{2} \)
37 \( 1 + 4.30T + 37T^{2} \)
41 \( 1 - 6.28T + 41T^{2} \)
43 \( 1 - 0.639T + 43T^{2} \)
47 \( 1 - 4.82T + 47T^{2} \)
53 \( 1 - 3.32T + 53T^{2} \)
59 \( 1 + 8.90T + 59T^{2} \)
61 \( 1 - 8.06T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 13.9T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 - 7.58T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 - 4.74T + 89T^{2} \)
97 \( 1 - 15.3T + 97T^{2} \)
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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

−9.014507219833987772454102698987, −8.848167797346294120093108735946, −7.34255557519508144525001258901, −6.45663555688282856725591836301, −5.90383405778042051701899428374, −5.23092620178189887859909950443, −4.34588800322975813728638896069, −3.27390286921320943536555635799, −2.40581902490591092453779849779, −1.88238114021611440476916185999, 1.88238114021611440476916185999, 2.40581902490591092453779849779, 3.27390286921320943536555635799, 4.34588800322975813728638896069, 5.23092620178189887859909950443, 5.90383405778042051701899428374, 6.45663555688282856725591836301, 7.34255557519508144525001258901, 8.848167797346294120093108735946, 9.014507219833987772454102698987

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