Properties

Label 2-1859-1.1-c1-0-50
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  − 0.776·2-s + 3.01·3-s − 1.39·4-s − 0.0164·5-s − 2.33·6-s − 0.793·7-s + 2.63·8-s + 6.06·9-s + 0.0127·10-s − 11-s − 4.20·12-s + 0.616·14-s − 0.0495·15-s + 0.747·16-s + 3.12·17-s − 4.70·18-s − 1.22·19-s + 0.0229·20-s − 2.38·21-s + 0.776·22-s + 4.26·23-s + 7.93·24-s − 4.99·25-s + 9.21·27-s + 1.10·28-s + 8.01·29-s + 0.0384·30-s + ⋯
L(s)  = 1  − 0.548·2-s + 1.73·3-s − 0.698·4-s − 0.00735·5-s − 0.953·6-s − 0.299·7-s + 0.932·8-s + 2.02·9-s + 0.00403·10-s − 0.301·11-s − 1.21·12-s + 0.164·14-s − 0.0127·15-s + 0.186·16-s + 0.758·17-s − 1.10·18-s − 0.281·19-s + 0.00514·20-s − 0.521·21-s + 0.165·22-s + 0.890·23-s + 1.62·24-s − 0.999·25-s + 1.77·27-s + 0.209·28-s + 1.48·29-s + 0.00702·30-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

\(L(1)\) \(\approx\) \(2.091542805\)
\(L(\frac12)\) \(\approx\) \(2.091542805\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.776T + 2T^{2} \)
3 \( 1 - 3.01T + 3T^{2} \)
5 \( 1 + 0.0164T + 5T^{2} \)
7 \( 1 + 0.793T + 7T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 + 1.22T + 19T^{2} \)
23 \( 1 - 4.26T + 23T^{2} \)
29 \( 1 - 8.01T + 29T^{2} \)
31 \( 1 + 2.49T + 31T^{2} \)
37 \( 1 - 8.88T + 37T^{2} \)
41 \( 1 - 8.43T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 12.2T + 47T^{2} \)
53 \( 1 - 4.56T + 53T^{2} \)
59 \( 1 + 0.265T + 59T^{2} \)
61 \( 1 + 4.49T + 61T^{2} \)
67 \( 1 + 5.31T + 67T^{2} \)
71 \( 1 - 13.1T + 71T^{2} \)
73 \( 1 - 11.5T + 73T^{2} \)
79 \( 1 + 9.35T + 79T^{2} \)
83 \( 1 - 4.81T + 83T^{2} \)
89 \( 1 - 1.66T + 89T^{2} \)
97 \( 1 + 8.35T + 97T^{2} \)
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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

−9.227356833363515596045933276796, −8.502626826863905738482539098381, −7.84046354248518948446845093759, −7.44797875457810097477987842510, −6.19775712498467604773597387346, −4.89699993907720020788983686784, −4.08170550512725666942254944895, −3.24454634110249998393006527189, −2.33682064660647311484667660059, −1.04602453798382507937057180977, 1.04602453798382507937057180977, 2.33682064660647311484667660059, 3.24454634110249998393006527189, 4.08170550512725666942254944895, 4.89699993907720020788983686784, 6.19775712498467604773597387346, 7.44797875457810097477987842510, 7.84046354248518948446845093759, 8.502626826863905738482539098381, 9.227356833363515596045933276796

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