Properties

Label 2-1859-1.1-c1-0-22
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  − 0.930·2-s + 2.53·3-s − 1.13·4-s − 3.38·5-s − 2.35·6-s − 3.38·7-s + 2.91·8-s + 3.41·9-s + 3.14·10-s + 11-s − 2.87·12-s + 3.14·14-s − 8.57·15-s − 0.441·16-s − 7.19·17-s − 3.17·18-s − 3.06·19-s + 3.84·20-s − 8.56·21-s − 0.930·22-s + 5.94·23-s + 7.38·24-s + 6.47·25-s + 1.05·27-s + 3.83·28-s + 5.43·29-s + 7.97·30-s + ⋯
L(s)  = 1  − 0.657·2-s + 1.46·3-s − 0.567·4-s − 1.51·5-s − 0.961·6-s − 1.27·7-s + 1.03·8-s + 1.13·9-s + 0.996·10-s + 0.301·11-s − 0.829·12-s + 0.840·14-s − 2.21·15-s − 0.110·16-s − 1.74·17-s − 0.748·18-s − 0.703·19-s + 0.859·20-s − 1.86·21-s − 0.198·22-s + 1.24·23-s + 1.50·24-s + 1.29·25-s + 0.202·27-s + 0.725·28-s + 1.00·29-s + 1.45·30-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

\(L(1)\) \(\approx\) \(0.8727727082\)
\(L(\frac12)\) \(\approx\) \(0.8727727082\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.930T + 2T^{2} \)
3 \( 1 - 2.53T + 3T^{2} \)
5 \( 1 + 3.38T + 5T^{2} \)
7 \( 1 + 3.38T + 7T^{2} \)
17 \( 1 + 7.19T + 17T^{2} \)
19 \( 1 + 3.06T + 19T^{2} \)
23 \( 1 - 5.94T + 23T^{2} \)
29 \( 1 - 5.43T + 29T^{2} \)
31 \( 1 - 9.06T + 31T^{2} \)
37 \( 1 + 4.87T + 37T^{2} \)
41 \( 1 - 0.859T + 41T^{2} \)
43 \( 1 - 5.81T + 43T^{2} \)
47 \( 1 - 8.69T + 47T^{2} \)
53 \( 1 + 9.91T + 53T^{2} \)
59 \( 1 - 7.38T + 59T^{2} \)
61 \( 1 - 5.88T + 61T^{2} \)
67 \( 1 + 1.27T + 67T^{2} \)
71 \( 1 - 13.2T + 71T^{2} \)
73 \( 1 - 1.40T + 73T^{2} \)
79 \( 1 - 4.47T + 79T^{2} \)
83 \( 1 + 2.06T + 83T^{2} \)
89 \( 1 + 0.472T + 89T^{2} \)
97 \( 1 - 6.51T + 97T^{2} \)
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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

−8.956962881750213630007729121611, −8.611047334874522744371202295190, −8.000743124991606465768786328133, −7.11432903552842620301083730953, −6.55581720014479125449850386470, −4.67380959429910374920024909639, −4.08826682628700658419743236583, −3.35599628203444778812573493878, −2.45152119873491141180222049736, −0.63497775362486230414964094083, 0.63497775362486230414964094083, 2.45152119873491141180222049736, 3.35599628203444778812573493878, 4.08826682628700658419743236583, 4.67380959429910374920024909639, 6.55581720014479125449850386470, 7.11432903552842620301083730953, 8.000743124991606465768786328133, 8.611047334874522744371202295190, 8.956962881750213630007729121611

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