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Properties

Label 2-287-41.37-c1-0-10
Degree $2$
Conductor $287$
Sign $0.961 + 0.273i$
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  + (−1.49 + 1.08i)2-s − 1.69·3-s + (0.443 − 1.36i)4-s + (−0.353 + 1.08i)5-s + (2.53 − 1.84i)6-s + (−0.809 − 0.587i)7-s + (−0.323 − 0.996i)8-s − 0.136·9-s + (−0.654 − 2.01i)10-s + (−0.204 − 0.630i)11-s + (−0.750 + 2.30i)12-s + (−0.827 + 0.600i)13-s + 1.85·14-s + (0.597 − 1.84i)15-s + (3.89 + 2.82i)16-s + (−2.30 − 7.09i)17-s + ⋯
L(s)  = 1  + (−1.06 + 0.770i)2-s − 0.976·3-s + (0.221 − 0.682i)4-s + (−0.158 + 0.486i)5-s + (1.03 − 0.752i)6-s + (−0.305 − 0.222i)7-s + (−0.114 − 0.352i)8-s − 0.0456·9-s + (−0.207 − 0.637i)10-s + (−0.0617 − 0.190i)11-s + (−0.216 + 0.666i)12-s + (−0.229 + 0.166i)13-s + 0.495·14-s + (0.154 − 0.475i)15-s + (0.973 + 0.706i)16-s + (−0.558 − 1.72i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.961 + 0.273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.961 + 0.273i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.961 + 0.273i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336601 - 0.0468790i\)
\(L(\frac12)\) \(\approx\) \(0.336601 - 0.0468790i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (5.93 - 2.39i)T \)
good2 \( 1 + (1.49 - 1.08i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + 1.69T + 3T^{2} \)
5 \( 1 + (0.353 - 1.08i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (0.204 + 0.630i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.827 - 0.600i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (2.30 + 7.09i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-5.62 - 4.08i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-4.26 + 3.09i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-0.436 + 1.34i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.72 + 5.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.54 + 4.76i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (-3.37 + 2.45i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (1.68 - 1.22i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.31 + 4.05i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-1.44 + 1.05i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (10.3 + 7.50i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-4.25 + 13.0i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.15 + 6.64i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 14.9T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 0.921T + 83T^{2} \)
89 \( 1 + (3.47 + 2.52i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.179 - 0.550i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.52714501322212367225608143154, −10.77264927119931389969911618846, −9.733868679932101498756763667052, −8.995897248181986472153028243545, −7.67569675882631609040163069622, −6.99670705946928616158314281670, −6.14244722395285002487247389632, −4.99692043627749483805404650649, −3.19029559956716139697291029612, −0.48146611353755604801941229016, 1.17698704867825443855768083230, 2.96792056402008039411385214227, 4.86869288222188536689189147946, 5.77539687772892274453370206904, 7.07321930147836128777326050960, 8.457727846815831175503889775874, 9.052693456089657424538826898858, 10.17522753556165191034795035344, 10.85825079806922624819395607524, 11.66083194407107127396006002609

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