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Properties

Label 2-287-41.9-c1-0-13
Degree $2$
Conductor $287$
Sign $0.525 + 0.850i$
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  − 0.124i·2-s + (−0.249 − 0.249i)3-s + 1.98·4-s − 1.52i·5-s + (−0.0310 + 0.0310i)6-s + (−0.707 − 0.707i)7-s − 0.495i·8-s − 2.87i·9-s − 0.190·10-s + (−0.279 − 0.279i)11-s + (−0.496 − 0.496i)12-s + (0.882 + 0.882i)13-s + (−0.0878 + 0.0878i)14-s + (−0.382 + 0.382i)15-s + 3.90·16-s + (−2.79 + 2.79i)17-s + ⋯
L(s)  = 1  − 0.0878i·2-s + (−0.144 − 0.144i)3-s + 0.992·4-s − 0.683i·5-s + (−0.0126 + 0.0126i)6-s + (−0.267 − 0.267i)7-s − 0.175i·8-s − 0.958i·9-s − 0.0600·10-s + (−0.0841 − 0.0841i)11-s + (−0.143 − 0.143i)12-s + (0.244 + 0.244i)13-s + (−0.0234 + 0.0234i)14-s + (−0.0986 + 0.0986i)15-s + 0.976·16-s + (−0.677 + 0.677i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27712 - 0.712169i\)
\(L(\frac12)\) \(\approx\) \(1.27712 - 0.712169i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.707 + 0.707i)T \)
41 \( 1 + (-5.39 - 3.45i)T \)
good2 \( 1 + 0.124iT - 2T^{2} \)
3 \( 1 + (0.249 + 0.249i)T + 3iT^{2} \)
5 \( 1 + 1.52iT - 5T^{2} \)
11 \( 1 + (0.279 + 0.279i)T + 11iT^{2} \)
13 \( 1 + (-0.882 - 0.882i)T + 13iT^{2} \)
17 \( 1 + (2.79 - 2.79i)T - 17iT^{2} \)
19 \( 1 + (-1.35 + 1.35i)T - 19iT^{2} \)
23 \( 1 - 2.18T + 23T^{2} \)
29 \( 1 + (1.85 + 1.85i)T + 29iT^{2} \)
31 \( 1 + 2.55T + 31T^{2} \)
37 \( 1 - 6.48T + 37T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + (7.41 - 7.41i)T - 47iT^{2} \)
53 \( 1 + (0.749 + 0.749i)T + 53iT^{2} \)
59 \( 1 + 9.10T + 59T^{2} \)
61 \( 1 - 3.36iT - 61T^{2} \)
67 \( 1 + (9.94 - 9.94i)T - 67iT^{2} \)
71 \( 1 + (-0.251 - 0.251i)T + 71iT^{2} \)
73 \( 1 - 7.56iT - 73T^{2} \)
79 \( 1 + (-6.30 - 6.30i)T + 79iT^{2} \)
83 \( 1 - 2.37T + 83T^{2} \)
89 \( 1 + (7.91 + 7.91i)T + 89iT^{2} \)
97 \( 1 + (-11.1 + 11.1i)T - 97iT^{2} \)
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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

−11.55237337106524110273100786818, −10.99663372711151395115792290923, −9.759384886800043147584457706833, −8.889353542722841299993728750955, −7.68132195694922176173019623858, −6.63153552296902559995360408948, −5.91596052621035792988657482437, −4.35033043012683311248458684731, −2.99997926318846003844902732683, −1.25338220916087647512398738490, 2.17493853842021848996218870746, 3.25209832971757214977618622008, 5.02732882121234068391347460142, 6.11712397719469432399070270014, 7.08798363119352052434311359420, 7.85067985353673977400391867875, 9.194812847843899271966650038549, 10.46272719591911664706235793035, 10.91869019095312412848994670829, 11.74663311515450337124491548998

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