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Properties

Label 2-287-7.2-c1-0-12
Degree $2$
Conductor $287$
Sign $0.605 - 0.795i$
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  + (0.651 + 1.12i)3-s + (1 + 1.73i)4-s + (1.15 − 1.99i)5-s + (−0.5 + 2.59i)7-s + (0.651 − 1.12i)9-s + (0.348 + 0.603i)11-s + (−1.30 + 2.25i)12-s + 1.30·13-s + 2.99·15-s + (−1.99 + 3.46i)16-s + (−3.45 − 5.98i)17-s + (−1.80 + 3.12i)19-s + 4.60·20-s + (−3.25 + 1.12i)21-s + (−0.802 + 1.39i)23-s + ⋯
L(s)  = 1  + (0.376 + 0.651i)3-s + (0.5 + 0.866i)4-s + (0.514 − 0.891i)5-s + (−0.188 + 0.981i)7-s + (0.217 − 0.376i)9-s + (0.105 + 0.182i)11-s + (−0.376 + 0.651i)12-s + 0.361·13-s + 0.774·15-s + (−0.499 + 0.866i)16-s + (−0.837 − 1.45i)17-s + (−0.413 + 0.716i)19-s + 1.02·20-s + (−0.710 + 0.246i)21-s + (−0.167 + 0.289i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.605 - 0.795i$
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: \((2,\ 287,\ (\ :1/2),\ 0.605 - 0.795i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.51515 + 0.751104i\)
\(L(\frac12)\) \(\approx\) \(1.51515 + 0.751104i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.5 - 2.59i)T \)
41 \( 1 + T \)
good2 \( 1 + (-1 - 1.73i)T^{2} \)
3 \( 1 + (-0.651 - 1.12i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.15 + 1.99i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.348 - 0.603i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.30T + 13T^{2} \)
17 \( 1 + (3.45 + 5.98i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.80 - 3.12i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.802 - 1.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 1.60T + 29T^{2} \)
31 \( 1 + (0.651 + 1.12i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.60 + 6.24i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 2.39T + 43T^{2} \)
47 \( 1 + (-4.5 + 7.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-0.454 - 0.786i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.80 + 6.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.60 + 6.24i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.302 + 0.524i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10.8T + 71T^{2} \)
73 \( 1 + (2.84 + 4.93i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.71 - 9.89i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 4.39T + 83T^{2} \)
89 \( 1 + (3.80 - 6.58i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 5.69T + 97T^{2} \)
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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

−12.08326743480749412550126853964, −11.16041877534561172371505849557, −9.738563541495721471790542768667, −9.076871988966826141726830898377, −8.463572576090920317636374546475, −7.10302288634510060576446297048, −5.93233671919917293997477353915, −4.68111596723156696194162622469, −3.49315128659335992345825347482, −2.17409286342795640919555006178, 1.53452887495229864297918084356, 2.71700601123590680600612959564, 4.42338803334240231018438504502, 6.14213393532285612177726148734, 6.66300408993220155842079487577, 7.53468988366182886751457448682, 8.793487035667235596291807078398, 10.26243526900362959244266983715, 10.52128783369452914481877669742, 11.37614503795906637077496506823

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