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Properties

Label 2-287-7.2-c1-0-18
Degree $2$
Conductor $287$
Sign $0.386 + 0.922i$
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  + (0.309 − 0.535i)2-s + (−0.5 − 0.866i)3-s + (0.809 + 1.40i)4-s + (0.5 − 0.866i)5-s − 0.618·6-s + (−2 − 1.73i)7-s + 2.23·8-s + (1 − 1.73i)9-s + (−0.309 − 0.535i)10-s + (−0.5 − 0.866i)11-s + (0.809 − 1.40i)12-s + 4.47·13-s + (−1.54 + 0.535i)14-s − 0.999·15-s + (−0.927 + 1.60i)16-s + (0.118 + 0.204i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.378i)2-s + (−0.288 − 0.499i)3-s + (0.404 + 0.700i)4-s + (0.223 − 0.387i)5-s − 0.252·6-s + (−0.755 − 0.654i)7-s + 0.790·8-s + (0.333 − 0.577i)9-s + (−0.0977 − 0.169i)10-s + (−0.150 − 0.261i)11-s + (0.233 − 0.404i)12-s + 1.24·13-s + (−0.412 + 0.143i)14-s − 0.258·15-s + (−0.231 + 0.401i)16-s + (0.0286 + 0.0495i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.386 + 0.922i$
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: \((2,\ 287,\ (\ :1/2),\ 0.386 + 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23390 - 0.820770i\)
\(L(\frac12)\) \(\approx\) \(1.23390 - 0.820770i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2 + 1.73i)T \)
41 \( 1 + T \)
good2 \( 1 + (-0.309 + 0.535i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.47T + 13T^{2} \)
17 \( 1 + (-0.118 - 0.204i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.73 + 6.47i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.35 - 4.07i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.47T + 29T^{2} \)
31 \( 1 + (2.11 + 3.66i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (2.73 - 4.73i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 2.47T + 43T^{2} \)
47 \( 1 + (2.73 - 4.73i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.88 - 4.99i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.35 - 4.07i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.263 - 0.457i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.97 - 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.52T + 71T^{2} \)
73 \( 1 + (-5.73 - 9.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.4T + 83T^{2} \)
89 \( 1 + (-4.88 + 8.45i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.472T + 97T^{2} \)
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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

−11.58863448208178144509214254084, −11.08448072087090382888086870923, −9.794643089299078825741633453552, −8.838861846142483031015722302022, −7.49882096107571855039872080087, −6.85215040429378768294383218778, −5.73928340159320297056935962984, −4.06654847480931455866256487292, −3.15200471474871487295236106604, −1.25856922677398735286318870674, 1.99729129393781161489128864507, 3.71065248310360537097622664703, 5.20757442594158331473375367004, 5.94496804984481610886579400464, 6.79496532720299156991386605132, 8.039446688044265073909879385384, 9.414185830738884225980505887323, 10.30284127447950246220442100595, 10.74401994515805580582174485608, 11.91223943646148600892052048104

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