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Properties

Label 2-287-7.2-c1-0-16
Degree $2$
Conductor $287$
Sign $0.788 - 0.614i$
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  + (0.451 − 0.781i)2-s + (1.66 + 2.87i)3-s + (0.592 + 1.02i)4-s + (0.469 − 0.812i)5-s + 3.00·6-s + (1.14 − 2.38i)7-s + 2.87·8-s + (−4.02 + 6.97i)9-s + (−0.423 − 0.733i)10-s + (−2.73 − 4.72i)11-s + (−1.97 + 3.41i)12-s − 2.52·13-s + (−1.34 − 1.97i)14-s + 3.12·15-s + (0.111 − 0.193i)16-s + (−3.35 − 5.81i)17-s + ⋯
L(s)  = 1  + (0.319 − 0.552i)2-s + (0.959 + 1.66i)3-s + (0.296 + 0.513i)4-s + (0.209 − 0.363i)5-s + 1.22·6-s + (0.433 − 0.901i)7-s + 1.01·8-s + (−1.34 + 2.32i)9-s + (−0.133 − 0.231i)10-s + (−0.823 − 1.42i)11-s + (−0.568 + 0.985i)12-s − 0.700·13-s + (−0.359 − 0.527i)14-s + 0.805·15-s + (0.0279 − 0.0483i)16-s + (−0.813 − 1.40i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.788 - 0.614i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.788 - 0.614i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.788 - 0.614i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04387 + 0.702461i\)
\(L(\frac12)\) \(\approx\) \(2.04387 + 0.702461i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-1.14 + 2.38i)T \)
41 \( 1 - T \)
good2 \( 1 + (-0.451 + 0.781i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-1.66 - 2.87i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.469 + 0.812i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.73 + 4.72i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 2.52T + 13T^{2} \)
17 \( 1 + (3.35 + 5.81i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.428 - 0.742i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.38 - 4.12i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.28T + 29T^{2} \)
31 \( 1 + (0.0596 + 0.103i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.90 + 3.30i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 1.64T + 43T^{2} \)
47 \( 1 + (2.36 - 4.10i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.12 + 1.94i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (2.22 + 3.85i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5.98 - 10.3i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.86 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 9.78T + 71T^{2} \)
73 \( 1 + (-1.85 - 3.21i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.40 + 2.43i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.41T + 83T^{2} \)
89 \( 1 + (-7.12 + 12.3i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 8.40T + 97T^{2} \)
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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

−11.46843773907129204609366950785, −10.97398004700969886372534332816, −10.16817006359198515831885010799, −9.203197417736133002536951794481, −8.232248792866045062013036235328, −7.46952379405104757874307289758, −5.24740665921628583025999151478, −4.47586069070094964338076626720, −3.44204247944417003532000380827, −2.55248151963276240899056812298, 1.97054781426045196618145504936, 2.43113551360471006006952198365, 4.77193549307714179228497234370, 6.20221193030763076050778246148, 6.72880592285676300848622592993, 7.74825788778927727871339417082, 8.393204233310541396692481702017, 9.687674007000924254726175965556, 10.82573511402167682219300755819, 12.27813383743224173192059698796

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