↑

Properties

Label 2-285-285.197-c1-0-27
Degree $2$
Conductor $285$
Sign $-0.484 + 0.874i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.863 + 0.231i)2-s + (−1.58 − 0.701i)3-s + (−1.03 − 0.600i)4-s + (1.45 + 1.69i)5-s + (−1.20 − 0.972i)6-s + (−2.61 − 2.61i)7-s + (−2.02 − 2.02i)8-s + (2.01 + 2.22i)9-s + (0.863 + 1.80i)10-s − 4.81i·11-s + (1.22 + 1.67i)12-s + (−0.748 − 2.79i)13-s + (−1.65 − 2.86i)14-s + (−1.11 − 3.70i)15-s + (−0.0792 − 0.137i)16-s + (−4.94 − 1.32i)17-s + ⋯
L(s)  = 1  + (0.610 + 0.163i)2-s + (−0.914 − 0.404i)3-s + (−0.519 − 0.300i)4-s + (0.650 + 0.759i)5-s + (−0.492 − 0.396i)6-s + (−0.990 − 0.990i)7-s + (−0.715 − 0.715i)8-s + (0.672 + 0.740i)9-s + (0.272 + 0.570i)10-s − 1.45i·11-s + (0.353 + 0.484i)12-s + (−0.207 − 0.774i)13-s + (−0.442 − 0.766i)14-s + (−0.287 − 0.957i)15-s + (−0.0198 − 0.0343i)16-s + (−1.19 − 0.321i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $-0.484 + 0.874i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ -0.484 + 0.874i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.387509 - 0.657557i\)
\(L(\frac12)\) \(\approx\) \(0.387509 - 0.657557i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.58 + 0.701i)T \)
5 \( 1 + (-1.45 - 1.69i)T \)
19 \( 1 + (3.95 - 1.82i)T \)
good2 \( 1 + (-0.863 - 0.231i)T + (1.73 + i)T^{2} \)
7 \( 1 + (2.61 + 2.61i)T + 7iT^{2} \)
11 \( 1 + 4.81iT - 11T^{2} \)
13 \( 1 + (0.748 + 2.79i)T + (-11.2 + 6.5i)T^{2} \)
17 \( 1 + (4.94 + 1.32i)T + (14.7 + 8.5i)T^{2} \)
23 \( 1 + (-3.94 + 1.05i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + (-0.781 + 1.35i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 6.98T + 31T^{2} \)
37 \( 1 + (-1.33 - 1.33i)T + 37iT^{2} \)
41 \( 1 + (0.965 - 0.557i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.60 - 0.698i)T + (37.2 + 21.5i)T^{2} \)
47 \( 1 + (1.32 + 4.92i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-10.5 + 2.82i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.76 - 3.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.0117 + 0.0203i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (9.55 - 2.55i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.65 + 0.954i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (7.22 + 1.93i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.90 - 2.83i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (10.1 + 10.1i)T + 83iT^{2} \)
89 \( 1 + (-6.13 + 10.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.85 + 6.90i)T + (-84.0 - 48.5i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.46444528968964996451764018739, −10.45567671116475835285154591730, −10.10002380075381704246818925545, −8.692701874120459136792459478787, −7.06197036951628235127290619236, −6.33594839484572659981396850348, −5.73143960418829603721199323104, −4.39002290110882821137662875761, −3.05342230907547649767177450895, −0.52015816948468108214731107985, 2.39635247541541801151534061420, 4.29757466520012385043456444989, 4.83596095926484627018089679529, 5.95192768380200750787065253257, 6.79905741479702260483247962962, 8.837788974005266641469145712713, 9.263449315568688224497538272713, 10.13382995797674368288403956511, 11.54411634261161542364633826288, 12.45731142252247704969566584464

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