Properties

Label 2-1859-1.1-c1-0-108
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  − 2.50·2-s + 2.28·3-s + 4.28·4-s + 1.22·5-s − 5.72·6-s − 0.778·7-s − 5.72·8-s + 2.22·9-s − 3.06·10-s − 11-s + 9.79·12-s + 1.95·14-s + 2.79·15-s + 5.79·16-s − 4.28·17-s − 5.57·18-s − 8.29·19-s + 5.23·20-s − 1.77·21-s + 2.50·22-s − 2.72·23-s − 13.0·24-s − 3.50·25-s − 1.77·27-s − 3.33·28-s + 7.36·29-s − 6.99·30-s + ⋯
L(s)  = 1  − 1.77·2-s + 1.31·3-s + 2.14·4-s + 0.546·5-s − 2.33·6-s − 0.294·7-s − 2.02·8-s + 0.740·9-s − 0.968·10-s − 0.301·11-s + 2.82·12-s + 0.521·14-s + 0.720·15-s + 1.44·16-s − 1.03·17-s − 1.31·18-s − 1.90·19-s + 1.17·20-s − 0.388·21-s + 0.534·22-s − 0.569·23-s − 2.67·24-s − 0.701·25-s − 0.342·27-s − 0.630·28-s + 1.36·29-s − 1.27·30-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: \(1\)
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 \( 1 + 2.50T + 2T^{2} \)
3 \( 1 - 2.28T + 3T^{2} \)
5 \( 1 - 1.22T + 5T^{2} \)
7 \( 1 + 0.778T + 7T^{2} \)
17 \( 1 + 4.28T + 17T^{2} \)
19 \( 1 + 8.29T + 19T^{2} \)
23 \( 1 + 2.72T + 23T^{2} \)
29 \( 1 - 7.36T + 29T^{2} \)
31 \( 1 - 1.44T + 31T^{2} \)
37 \( 1 - 0.792T + 37T^{2} \)
41 \( 1 + 9.74T + 41T^{2} \)
43 \( 1 + 9.67T + 43T^{2} \)
47 \( 1 - 3.38T + 47T^{2} \)
53 \( 1 - 1.79T + 53T^{2} \)
59 \( 1 + 5.20T + 59T^{2} \)
61 \( 1 + 14.5T + 61T^{2} \)
67 \( 1 + 6.06T + 67T^{2} \)
71 \( 1 - 6.50T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 - 13.5T + 79T^{2} \)
83 \( 1 - 13.3T + 83T^{2} \)
89 \( 1 - 4.77T + 89T^{2} \)
97 \( 1 - 4.14T + 97T^{2} \)
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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

−8.833033730251007430788972250219, −8.283346070774721540305903936990, −7.77679022687989322394908832577, −6.61648830143272313699901820048, −6.28541197371515858869159902160, −4.59852399366341908751771238867, −3.29501434698583934126098022932, −2.28286188556339518328089312824, −1.85121565698763246165816699171, 0, 1.85121565698763246165816699171, 2.28286188556339518328089312824, 3.29501434698583934126098022932, 4.59852399366341908751771238867, 6.28541197371515858869159902160, 6.61648830143272313699901820048, 7.77679022687989322394908832577, 8.283346070774721540305903936990, 8.833033730251007430788972250219

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