Properties

Label 2-287-287.32-c1-0-15
Degree $2$
Conductor $287$
Sign $0.447 + 0.894i$
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  + (−0.866 + 0.5i)2-s + (−0.5 − 1.86i)3-s + (−0.500 + 0.866i)4-s + (−1.5 + 0.866i)5-s + (1.36 + 1.36i)6-s + (2.5 + 0.866i)7-s − 3i·8-s + (−0.633 + 0.366i)9-s + (0.866 − 1.5i)10-s + (−1.13 − 4.23i)11-s + (1.86 + 0.500i)12-s + (−1.73 + 1.73i)13-s + (−2.59 + 0.500i)14-s + (2.36 + 2.36i)15-s + (0.500 + 0.866i)16-s + (6.46 − 1.73i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.288 − 1.07i)3-s + (−0.250 + 0.433i)4-s + (−0.670 + 0.387i)5-s + (0.557 + 0.557i)6-s + (0.944 + 0.327i)7-s − 1.06i·8-s + (−0.211 + 0.122i)9-s + (0.273 − 0.474i)10-s + (−0.341 − 1.27i)11-s + (0.538 + 0.144i)12-s + (−0.480 + 0.480i)13-s + (−0.694 + 0.133i)14-s + (0.610 + 0.610i)15-s + (0.125 + 0.216i)16-s + (1.56 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.447 + 0.894i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (32, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.586261 - 0.362137i\)
\(L(\frac12)\) \(\approx\) \(0.586261 - 0.362137i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.5 - 0.866i)T \)
41 \( 1 + (-4 - 5i)T \)
good2 \( 1 + (0.866 - 0.5i)T + (1 - 1.73i)T^{2} \)
3 \( 1 + (0.5 + 1.86i)T + (-2.59 + 1.5i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (1.13 + 4.23i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (1.73 - 1.73i)T - 13iT^{2} \)
17 \( 1 + (-6.46 + 1.73i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-1.40 + 5.23i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.73 + 6.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.73 + 6.73i)T - 29iT^{2} \)
31 \( 1 + (0.366 - 0.633i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.267 + 0.464i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 5.46iT - 43T^{2} \)
47 \( 1 + (1.63 - 6.09i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-0.366 - 1.36i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.36 - 9.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (6.06 - 3.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-2.36 + 0.633i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-6.63 + 6.63i)T - 71iT^{2} \)
73 \( 1 + (3.92 + 2.26i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-10.3 - 2.76i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 - 0.732T + 83T^{2} \)
89 \( 1 + (6.83 + 1.83i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (5.46 + 5.46i)T + 97iT^{2} \)
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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

−11.95434727341129138128628095659, −10.88610915345473683383838910983, −9.549120531605300005956522106739, −8.326900051272817066089826906977, −7.83581135101102614985336308868, −7.09818640262132342470421212234, −5.97889491626366333250614007184, −4.44032515007122986701165539456, −2.86680015480360249496612369712, −0.73395917718888594622106665778, 1.55642298247629892690490389631, 3.84005659900998156308865243467, 4.89562546218098390002880026239, 5.44704789319198853249801868324, 7.67899981794446296763254117928, 8.105951383870364837114297515224, 9.578291529219052329323295133800, 10.11162570006157599471553525786, 10.67225329160758405280215518966, 11.81151233532647929085609757097

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