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Properties

Label 2-287-41.16-c1-0-12
Degree $2$
Conductor $287$
Sign $0.996 + 0.0852i$
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  + (0.223 − 0.688i)2-s + 1.30·3-s + (1.19 + 0.867i)4-s + (1.13 + 0.823i)5-s + (0.292 − 0.900i)6-s + (0.309 + 0.951i)7-s + (2.03 − 1.47i)8-s − 1.29·9-s + (0.821 − 0.596i)10-s + (−3.82 + 2.77i)11-s + (1.55 + 1.13i)12-s + (1.16 − 3.58i)13-s + 0.724·14-s + (1.48 + 1.07i)15-s + (0.348 + 1.07i)16-s + (−4.10 + 2.98i)17-s + ⋯
L(s)  = 1  + (0.158 − 0.487i)2-s + 0.754·3-s + (0.596 + 0.433i)4-s + (0.507 + 0.368i)5-s + (0.119 − 0.367i)6-s + (0.116 + 0.359i)7-s + (0.720 − 0.523i)8-s − 0.430·9-s + (0.259 − 0.188i)10-s + (−1.15 + 0.837i)11-s + (0.450 + 0.327i)12-s + (0.323 − 0.995i)13-s + 0.193·14-s + (0.382 + 0.277i)15-s + (0.0871 + 0.268i)16-s + (−0.994 + 0.722i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 + 0.0852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 + 0.0852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.996 + 0.0852i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.996 + 0.0852i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.01509 - 0.0860860i\)
\(L(\frac12)\) \(\approx\) \(2.01509 - 0.0860860i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.05 - 6.06i)T \)
good2 \( 1 + (-0.223 + 0.688i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 - 1.30T + 3T^{2} \)
5 \( 1 + (-1.13 - 0.823i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (3.82 - 2.77i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-1.16 + 3.58i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (4.10 - 2.98i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.533 + 1.64i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (-2.88 + 8.87i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-1.31 - 0.956i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.88 + 3.55i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (6.39 + 4.64i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-2.36 + 7.28i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (0.167 - 0.516i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-4.64 - 3.37i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.97 - 12.2i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.76 - 8.51i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-6.71 - 4.87i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (7.47 - 5.43i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 3.43T + 73T^{2} \)
79 \( 1 + 6.51T + 79T^{2} \)
83 \( 1 + 12.6T + 83T^{2} \)
89 \( 1 + (0.836 + 2.57i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-12.9 - 9.37i)T + (29.9 + 92.2i)T^{2} \)
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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

−11.85918531176340653992551106012, −10.57345828027438275076548474875, −10.39104598800766695581736727795, −8.780587488906985381754377782454, −8.104496204880297275006462111502, −7.00586261936718746458606262004, −5.85269264181971067615513977949, −4.32937089138703572374667769247, −2.68459787841937293464561631663, −2.42570643473320160177577340037, 1.82964581855008884830734776201, 3.16366200403841161525516530135, 4.94455711853635652376369690468, 5.81266512999534712310583724794, 6.95182512586485577586606512016, 7.954773546060973106793929478652, 8.866388391278950640369574447456, 9.819202653269568981674890581821, 11.02656304545555788740670345679, 11.53887315009106040789494501058

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