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Properties

Label 2-1860-31.25-c1-0-1
Degree $2$
Conductor $1860$
Sign $-0.993 + 0.117i$
Analytic cond. $14.8521$
Root an. cond. $3.85385$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + (1.28 + 2.22i)7-s + (−0.499 + 0.866i)9-s + (−1.07 + 1.85i)11-s + (−2.98 + 5.17i)13-s − 0.999·15-s + (−2.17 − 3.75i)17-s + (−1.98 − 3.44i)19-s + (−1.28 + 2.22i)21-s − 7.65·23-s + (−0.499 − 0.866i)25-s − 0.999·27-s + 0.639·29-s + (5.34 + 1.55i)31-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.223 + 0.387i)5-s + (0.485 + 0.841i)7-s + (−0.166 + 0.288i)9-s + (−0.322 + 0.558i)11-s + (−0.829 + 1.43i)13-s − 0.258·15-s + (−0.526 − 0.911i)17-s + (−0.456 − 0.790i)19-s + (−0.280 + 0.485i)21-s − 1.59·23-s + (−0.0999 − 0.173i)25-s − 0.192·27-s + 0.118·29-s + (0.960 + 0.279i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.117i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $-0.993 + 0.117i$
Analytic conductor: \(14.8521\)
Root analytic conductor: \(3.85385\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1860} (1141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1860,\ (\ :1/2),\ -0.993 + 0.117i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8525637356\)
\(L(\frac12)\) \(\approx\) \(0.8525637356\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-5.34 - 1.55i)T \)
good7 \( 1 + (-1.28 - 2.22i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1.07 - 1.85i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.98 - 5.17i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (2.17 + 3.75i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.98 + 3.44i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 7.65T + 23T^{2} \)
29 \( 1 - 0.639T + 29T^{2} \)
37 \( 1 + (-0.285 - 0.493i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.402 + 0.697i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.60 + 6.24i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.229T + 47T^{2} \)
53 \( 1 + (-5.76 + 9.97i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2.25 - 3.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 0.358T + 61T^{2} \)
67 \( 1 + (2.12 - 3.67i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (5.55 - 9.62i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (5.50 - 9.53i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-7.60 - 13.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.33 + 9.23i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 3.50T + 89T^{2} \)
97 \( 1 + 8.22T + 97T^{2} \)
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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

−9.685494394870552713699679433393, −8.827534270009844718693693676490, −8.273565582868188293688413418384, −7.19665083874377350281105148985, −6.66552878565241679589456445233, −5.43290915299762903374542574894, −4.66995539848554057423361956342, −4.02081308250926469001920039514, −2.54055350111536547398424441917, −2.15242416290783036185774122517, 0.28269178999237350822449855495, 1.55098464215452802327668712984, 2.74188224781168589327233427351, 3.83907902207692681530262088090, 4.61802428725187023102212513794, 5.71879184859956311991770715345, 6.38936560864270367071581327276, 7.65736775485054106281226527725, 7.943926308408898354125868249201, 8.481807103690247203631527095355

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