Properties

Label 2-287-41.18-c1-0-0
Degree $2$
Conductor $287$
Sign $0.253 - 0.967i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.508 − 1.56i)2-s − 0.349·3-s + (−0.574 + 0.417i)4-s + (−3.08 + 2.24i)5-s + (0.177 + 0.547i)6-s + (0.309 − 0.951i)7-s + (−1.71 − 1.24i)8-s − 2.87·9-s + (5.07 + 3.68i)10-s + (3.11 + 2.26i)11-s + (0.200 − 0.145i)12-s + (1.80 + 5.56i)13-s − 1.64·14-s + (1.07 − 0.783i)15-s + (−1.51 + 4.67i)16-s + (−5.63 − 4.09i)17-s + ⋯
L(s)  = 1  + (−0.359 − 1.10i)2-s − 0.201·3-s + (−0.287 + 0.208i)4-s + (−1.37 + 1.00i)5-s + (0.0725 + 0.223i)6-s + (0.116 − 0.359i)7-s + (−0.607 − 0.441i)8-s − 0.959·9-s + (1.60 + 1.16i)10-s + (0.939 + 0.682i)11-s + (0.0580 − 0.0421i)12-s + (0.501 + 1.54i)13-s − 0.440·14-s + (0.278 − 0.202i)15-s + (−0.379 + 1.16i)16-s + (−1.36 − 0.992i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.253 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.253 - 0.967i$
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: \((2,\ 287,\ (\ :1/2),\ 0.253 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.243248 + 0.187654i\)
\(L(\frac12)\) \(\approx\) \(0.243248 + 0.187654i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.02 + 3.97i)T \)
good2 \( 1 + (0.508 + 1.56i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 0.349T + 3T^{2} \)
5 \( 1 + (3.08 - 2.24i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-3.11 - 2.26i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.80 - 5.56i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.63 + 4.09i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.51 - 7.74i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.215 + 0.663i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (1.00 - 0.730i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.518 + 0.376i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.19 - 1.59i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (2.63 + 8.09i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.485 - 1.49i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (5.32 - 3.87i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (0.190 + 0.586i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-1.32 + 4.08i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (4.02 - 2.92i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (0.381 + 0.277i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 7.57T + 73T^{2} \)
79 \( 1 + 3.79T + 79T^{2} \)
83 \( 1 - 1.48T + 83T^{2} \)
89 \( 1 + (3.66 - 11.2i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (1.54 - 1.12i)T + (29.9 - 92.2i)T^{2} \)
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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

−11.79880825477854462354666688436, −11.17688188407692543370706112156, −10.54928887669383512710477868736, −9.314607846346622908678051369175, −8.428918518532920536148130966201, −6.99179208356573339640650355352, −6.44832739174750134308702813329, −4.24142325595401080855060025430, −3.52939528449649193613240953141, −2.03410166362919147336308156366, 0.25321385851632409372812975677, 3.21417969111251883355152259925, 4.68329753155057906289484798384, 5.78713958195862999772789383543, 6.70392082278415167631450620635, 8.036998974429297213646612951369, 8.568202164418871541723674619145, 8.960587957838951784758408756455, 11.17101910918453062351642464450, 11.38125669686404527395725748303

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