↑

Properties

Label 2-287-41.18-c1-0-10
Degree $2$
Conductor $287$
Sign $0.530 - 0.847i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
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  + (0.450 + 1.38i)2-s + 0.560·3-s + (−0.103 + 0.0749i)4-s + (0.738 − 0.536i)5-s + (0.252 + 0.777i)6-s + (0.309 − 0.951i)7-s + (2.20 + 1.60i)8-s − 2.68·9-s + (1.07 + 0.782i)10-s + (3.83 + 2.78i)11-s + (−0.0578 + 0.0420i)12-s + (−0.942 − 2.90i)13-s + 1.45·14-s + (0.413 − 0.300i)15-s + (−1.30 + 4.03i)16-s + (2.35 + 1.71i)17-s + ⋯
L(s)  = 1  + (0.318 + 0.980i)2-s + 0.323·3-s + (−0.0516 + 0.0374i)4-s + (0.330 − 0.239i)5-s + (0.103 + 0.317i)6-s + (0.116 − 0.359i)7-s + (0.781 + 0.567i)8-s − 0.895·9-s + (0.340 + 0.247i)10-s + (1.15 + 0.840i)11-s + (−0.0166 + 0.0121i)12-s + (−0.261 − 0.804i)13-s + 0.389·14-s + (0.106 − 0.0776i)15-s + (−0.327 + 1.00i)16-s + (0.571 + 0.415i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.530 - 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.530 - 0.847i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.530 - 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65712 + 0.917761i\)
\(L(\frac12)\) \(\approx\) \(1.65712 + 0.917761i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (3.60 + 5.28i)T \)
good2 \( 1 + (-0.450 - 1.38i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 - 0.560T + 3T^{2} \)
5 \( 1 + (-0.738 + 0.536i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-3.83 - 2.78i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.942 + 2.90i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-2.35 - 1.71i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.344 - 1.06i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.29 + 7.07i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.29 - 3.12i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (4.81 + 3.50i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (1.13 - 0.826i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-0.0367 - 0.113i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (1.55 + 4.77i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.150 - 0.109i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.53 - 4.73i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.40 + 7.39i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (3.78 - 2.75i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (2.11 + 1.53i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 15.8T + 73T^{2} \)
79 \( 1 + 9.40T + 79T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 + (-4.17 + 12.8i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (7.55 - 5.48i)T + (29.9 - 92.2i)T^{2} \)
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

−12.10026260542898359493551017466, −10.98568555485454090246368809299, −10.01636703369860400658800648578, −8.892084858633788373184720879126, −7.942944146007686904505929842223, −7.04125966699612869439139077932, −5.99529173178648619956672992825, −5.14773413185571615041633174828, −3.79654770556979013376587324843, −1.92959744318620211014323717020, 1.77746955925203666384775917482, 3.01355669524998950838574300339, 3.94718907942076384380358056862, 5.54454663287523295036889438840, 6.66677323566122091735405278503, 7.906104132402196141652229775877, 9.108134554781202871278122046327, 9.780009141618937892680523783430, 11.16491637553444102091721629916, 11.56440510062735373870176622177

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