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Properties

Label 2-287-287.214-c1-0-7
Degree $2$
Conductor $287$
Sign $0.209 - 0.977i$
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 + 0.133i)3-s + (−0.500 + 0.866i)4-s + (−1.5 + 0.866i)5-s + (−0.366 + 0.366i)6-s + (2.5 + 0.866i)7-s + 3i·8-s + (−2.36 + 1.36i)9-s + (−0.866 + 1.5i)10-s + (−2.86 + 0.767i)11-s + (0.133 − 0.5i)12-s + (1.73 + 1.73i)13-s + (2.59 − 0.500i)14-s + (0.633 − 0.633i)15-s + (0.500 + 0.866i)16-s + (−0.464 − 1.73i)17-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (−0.288 + 0.0773i)3-s + (−0.250 + 0.433i)4-s + (−0.670 + 0.387i)5-s + (−0.149 + 0.149i)6-s + (0.944 + 0.327i)7-s + 1.06i·8-s + (−0.788 + 0.455i)9-s + (−0.273 + 0.474i)10-s + (−0.864 + 0.231i)11-s + (0.0386 − 0.144i)12-s + (0.480 + 0.480i)13-s + (0.694 − 0.133i)14-s + (0.163 − 0.163i)15-s + (0.125 + 0.216i)16-s + (−0.112 − 0.420i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.209 - 0.977i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (214, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951824 + 0.769842i\)
\(L(\frac12)\) \(\approx\) \(0.951824 + 0.769842i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.133i)T + (2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.86 - 0.767i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.73 - 1.73i)T + 13iT^{2} \)
17 \( 1 + (0.464 + 1.73i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-6.59 - 1.76i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.267 + 0.464i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.26 - 3.26i)T + 29iT^{2} \)
31 \( 1 + (-1.36 + 2.36i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.73 + 6.46i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 + (3.36 + 0.901i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.36 - 0.366i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (3.63 - 6.29i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-6.06 + 3.5i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.633 - 2.36i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-8.36 - 8.36i)T + 71iT^{2} \)
73 \( 1 + (-9.92 - 5.73i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.66 + 6.23i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + 2.73T + 83T^{2} \)
89 \( 1 + (-1.83 + 6.83i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.46 + 1.46i)T - 97iT^{2} \)
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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

−11.73432376867499974803704671838, −11.47304707139600852462359864020, −10.59022038089978850805838967890, −9.002405671832289853757750639202, −8.067554585563199438255799587719, −7.41194891929719928575938173058, −5.56108369038254359687301009321, −4.90857876017896760883263956844, −3.63655365298382858686317214009, −2.43461356258306072715760563754, 0.830326953249869851937098281622, 3.36099102996729294592844219173, 4.68916694910153168880059551447, 5.38147013761258229882730023092, 6.42136846021177150235846990349, 7.79969975409364441022558859394, 8.491974328731759750936061249018, 9.807458886230334990872532283305, 10.88079244843709402505105699836, 11.64741251920843358554469534920

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