Properties

Label 2-1860-31.25-c1-0-10
Degree $2$
Conductor $1860$
Sign $0.695 + 0.718i$
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 + (−0.499 + 0.866i)9-s + (2 − 3.46i)11-s + (1.5 − 2.59i)13-s + 0.999·15-s + (2 + 3.46i)17-s + (2.5 + 4.33i)19-s − 2·23-s + (−0.499 − 0.866i)25-s + 0.999·27-s − 6·29-s + (2 + 5.19i)31-s − 3.99·33-s + (−3.5 − 6.06i)37-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.603 − 1.04i)11-s + (0.416 − 0.720i)13-s + 0.258·15-s + (0.485 + 0.840i)17-s + (0.573 + 0.993i)19-s − 0.417·23-s + (−0.0999 − 0.173i)25-s + 0.192·27-s − 1.11·29-s + (0.359 + 0.933i)31-s − 0.696·33-s + (−0.575 − 0.996i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 + 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1860\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 31\)
Sign: $0.695 + 0.718i$
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: \((2,\ 1860,\ (\ :1/2),\ 0.695 + 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.520803684\)
\(L(\frac12)\) \(\approx\) \(1.520803684\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2 - 5.19i)T \)
good7 \( 1 + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1.5 + 2.59i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-2 - 3.46i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.5 - 4.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + 2T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
37 \( 1 + (3.5 + 6.06i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4 + 6.92i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 + (-2 + 3.46i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 10T + 61T^{2} \)
67 \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-3.5 + 6.06i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6 - 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1 + 1.73i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 - 4T + 89T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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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

−8.968424243209705889904749610090, −8.255384381760572554457279568417, −7.60736199192867486158505449666, −6.74462600671270894468855301098, −5.82380071692728006988432533841, −5.49660240013772622882155092793, −3.83188088838056662237899193439, −3.41215057028045922248812204269, −1.97347303620719470969677829244, −0.75406109417466985950559733146, 1.02065579828521175394748948964, 2.39791983285820431754999980172, 3.69981546431692415802750847098, 4.44022214496352150296741997319, 5.13001051292100262002988774942, 6.10915920228932421483499688273, 7.02109606437611604914713503569, 7.66721601174035139881331849444, 8.777576987284006642367863605650, 9.458845294266984180729975357966

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