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Properties

Label 2-287-41.40-c1-0-14
Degree $2$
Conductor $287$
Sign $-0.976 + 0.215i$
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  − 2.72·2-s − 1.72i·3-s + 5.44·4-s − 1.58·5-s + 4.71i·6-s − i·7-s − 9.39·8-s + 0.0134·9-s + 4.33·10-s − 6.33i·11-s − 9.40i·12-s + 2.36i·13-s + 2.72i·14-s + 2.74i·15-s + 14.7·16-s + 3.29i·17-s + ⋯
L(s)  = 1  − 1.92·2-s − 0.997i·3-s + 2.72·4-s − 0.710·5-s + 1.92i·6-s − 0.377i·7-s − 3.32·8-s + 0.00448·9-s + 1.37·10-s − 1.90i·11-s − 2.71i·12-s + 0.654i·13-s + 0.729i·14-s + 0.708i·15-s + 3.68·16-s + 0.799i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 + 0.215i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.976 + 0.215i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.976 + 0.215i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0339893 - 0.311706i\)
\(L(\frac12)\) \(\approx\) \(0.0339893 - 0.311706i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + iT \)
41 \( 1 + (-1.38 - 6.25i)T \)
good2 \( 1 + 2.72T + 2T^{2} \)
3 \( 1 + 1.72iT - 3T^{2} \)
5 \( 1 + 1.58T + 5T^{2} \)
11 \( 1 + 6.33iT - 11T^{2} \)
13 \( 1 - 2.36iT - 13T^{2} \)
17 \( 1 - 3.29iT - 17T^{2} \)
19 \( 1 + 3.34iT - 19T^{2} \)
23 \( 1 + 2.17T + 23T^{2} \)
29 \( 1 + 1.97iT - 29T^{2} \)
31 \( 1 + 7.04T + 31T^{2} \)
37 \( 1 + 6.69T + 37T^{2} \)
43 \( 1 + 6.85T + 43T^{2} \)
47 \( 1 - 4.19iT - 47T^{2} \)
53 \( 1 + 3.14iT - 53T^{2} \)
59 \( 1 + 7.58T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 14.0iT - 67T^{2} \)
71 \( 1 - 1.84iT - 71T^{2} \)
73 \( 1 - 6.52T + 73T^{2} \)
79 \( 1 + 6.15iT - 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 + 10.2iT - 89T^{2} \)
97 \( 1 + 9.34iT - 97T^{2} \)
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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

−11.29380239355265183169736370428, −10.45707760744312261673125954940, −9.243080196734656119700229574606, −8.311976626932674740510558906571, −7.81179251904461860098099753869, −6.84183951435009575461515756665, −6.10677248762859935231115470868, −3.46566316293367108496171668629, −1.79465922670144804092546012879, −0.42267841068974557955089973913, 2.00501447548063520102464617320, 3.68389976477465218589082816982, 5.29531555347809117192155158057, 7.00355854679962600095030364508, 7.58997730012184394654402778951, 8.645877791263890538820312125646, 9.625898922590103867437675522565, 10.04127485593891039792563409517, 10.86207400783554465383706419989, 11.92553738573701439670992365258

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