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Properties

Label 2-287-41.18-c1-0-13
Degree $2$
Conductor $287$
Sign $0.134 - 0.990i$
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  + (0.567 + 1.74i)2-s + 2.49·3-s + (−1.11 + 0.807i)4-s + (0.226 − 0.164i)5-s + (1.41 + 4.36i)6-s + (−0.309 + 0.951i)7-s + (0.931 + 0.676i)8-s + 3.23·9-s + (0.415 + 0.301i)10-s + (−5.12 − 3.72i)11-s + (−2.77 + 2.01i)12-s + (−1.81 − 5.59i)13-s − 1.83·14-s + (0.564 − 0.410i)15-s + (−1.50 + 4.62i)16-s + (3.18 + 2.31i)17-s + ⋯
L(s)  = 1  + (0.401 + 1.23i)2-s + 1.44·3-s + (−0.555 + 0.403i)4-s + (0.101 − 0.0734i)5-s + (0.578 + 1.77i)6-s + (−0.116 + 0.359i)7-s + (0.329 + 0.239i)8-s + 1.07·9-s + (0.131 + 0.0954i)10-s + (−1.54 − 1.12i)11-s + (−0.800 + 0.581i)12-s + (−0.504 − 1.55i)13-s − 0.490·14-s + (0.145 − 0.105i)15-s + (−0.375 + 1.15i)16-s + (0.771 + 0.560i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.134 - 0.990i$
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: \((2,\ 287,\ (\ :1/2),\ 0.134 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75607 + 1.53449i\)
\(L(\frac12)\) \(\approx\) \(1.75607 + 1.53449i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.46 - 3.33i)T \)
good2 \( 1 + (-0.567 - 1.74i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 - 2.49T + 3T^{2} \)
5 \( 1 + (-0.226 + 0.164i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (5.12 + 3.72i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.81 + 5.59i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.18 - 2.31i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (1.02 - 3.14i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.831 + 2.56i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-0.762 + 0.553i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.60 + 1.89i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.508 - 0.369i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (2.72 + 8.39i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-3.81 - 11.7i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.7 + 8.55i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.78 + 5.49i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.38 - 4.25i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.529 + 0.384i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-7.12 - 5.17i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 9.09T + 73T^{2} \)
79 \( 1 - 10.7T + 79T^{2} \)
83 \( 1 - 8.11T + 83T^{2} \)
89 \( 1 + (1.66 - 5.11i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (6.07 - 4.41i)T + (29.9 - 92.2i)T^{2} \)
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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

−12.65344567542751466015638330866, −10.82676804120056089895585997089, −9.998660593522393797751172892466, −8.630476785899329718746188104537, −7.993256734750045548687198384611, −7.57960976687499164040668824887, −5.87898115074044138240947787071, −5.32895800904617830621990735121, −3.57905639214657703115672514193, −2.49421708743906937714685959403, 2.06524993165376934510375348579, 2.69523860129877277008355090063, 3.95653463431025313641143646892, 4.89690821356828096143781918596, 7.12583727715497793356569703481, 7.68291777691119285341675901450, 9.103691484738419142141589855655, 9.847252301952654821208474687136, 10.51753905451946407696273217874, 11.77052898320416509054572146984

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