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Properties

Label 2-287-41.37-c1-0-19
Degree $2$
Conductor $287$
Sign $0.383 + 0.923i$
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  + (2.00 − 1.45i)2-s + 1.19·3-s + (1.28 − 3.96i)4-s + (−0.717 + 2.20i)5-s + (2.40 − 1.74i)6-s + (0.809 + 0.587i)7-s + (−1.66 − 5.11i)8-s − 1.56·9-s + (1.78 + 5.48i)10-s + (−1.22 − 3.76i)11-s + (1.54 − 4.74i)12-s + (−0.257 + 0.186i)13-s + 2.48·14-s + (−0.857 + 2.64i)15-s + (−4.07 − 2.95i)16-s + (1.63 + 5.03i)17-s + ⋯
L(s)  = 1  + (1.42 − 1.03i)2-s + 0.690·3-s + (0.643 − 1.98i)4-s + (−0.320 + 0.987i)5-s + (0.981 − 0.712i)6-s + (0.305 + 0.222i)7-s + (−0.588 − 1.81i)8-s − 0.523·9-s + (0.563 + 1.73i)10-s + (−0.368 − 1.13i)11-s + (0.444 − 1.36i)12-s + (−0.0713 + 0.0518i)13-s + 0.663·14-s + (−0.221 + 0.681i)15-s + (−1.01 − 0.739i)16-s + (0.397 + 1.22i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.383 + 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.383 + 0.923i$
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: \((2,\ 287,\ (\ :1/2),\ 0.383 + 0.923i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.43667 - 1.62624i\)
\(L(\frac12)\) \(\approx\) \(2.43667 - 1.62624i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-5.56 - 3.16i)T \)
good2 \( 1 + (-2.00 + 1.45i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 - 1.19T + 3T^{2} \)
5 \( 1 + (0.717 - 2.20i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (1.22 + 3.76i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.257 - 0.186i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.63 - 5.03i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.69 + 1.96i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (2.73 - 1.98i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (2.06 - 6.35i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (1.07 + 3.29i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.90 + 8.94i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (6.87 - 4.99i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-4.88 + 3.55i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.37 + 7.29i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.60 + 3.34i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-4.05 - 2.94i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-1.76 + 5.41i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.74 + 8.46i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + 10.1T + 73T^{2} \)
79 \( 1 - 8.26T + 79T^{2} \)
83 \( 1 + 12.7T + 83T^{2} \)
89 \( 1 + (-2.86 - 2.08i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.67 + 5.15i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.46405076847349022503193926248, −11.09842533699620281039119057179, −10.28340583133223219751051872643, −8.845795086380716345934276040814, −7.78378507849764746059650139830, −6.25171987072201154476872499597, −5.44048185260412830538736937888, −3.86304528891381792641808154639, −3.18052185986845273054024116492, −2.17389936973314199521343635812, 2.60320105302855697090456621062, 4.06930431974022863562541662096, 4.80372594899133961938875871161, 5.76116491148676778199106865533, 7.15181121940113834342901523247, 7.918956511438605511378070853829, 8.660681107941368636744019077328, 9.950273223269808156406402208012, 11.67311557100095717542616764755, 12.30169797192786111495053117674

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