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Properties

Label 2-287-7.4-c1-0-15
Degree $2$
Conductor $287$
Sign $0.658 - 0.752i$
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  + (0.834 + 1.44i)2-s + (0.106 − 0.184i)3-s + (−0.392 + 0.680i)4-s + (−0.501 − 0.868i)5-s + 0.356·6-s + (2.44 − 1.01i)7-s + 2.02·8-s + (1.47 + 2.55i)9-s + (0.837 − 1.45i)10-s + (1.75 − 3.04i)11-s + (0.0838 + 0.145i)12-s − 5.76·13-s + (3.50 + 2.69i)14-s − 0.214·15-s + (2.47 + 4.29i)16-s + (−2.43 + 4.21i)17-s + ⋯
L(s)  = 1  + (0.590 + 1.02i)2-s + (0.0616 − 0.106i)3-s + (−0.196 + 0.340i)4-s + (−0.224 − 0.388i)5-s + 0.145·6-s + (0.924 − 0.382i)7-s + 0.716·8-s + (0.492 + 0.852i)9-s + (0.264 − 0.458i)10-s + (0.530 − 0.917i)11-s + (0.0242 + 0.0419i)12-s − 1.59·13-s + (0.935 + 0.719i)14-s − 0.0553·15-s + (0.619 + 1.07i)16-s + (−0.590 + 1.02i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.658 - 0.752i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (165, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.658 - 0.752i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76779 + 0.802479i\)
\(L(\frac12)\) \(\approx\) \(1.76779 + 0.802479i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.44 + 1.01i)T \)
41 \( 1 - T \)
good2 \( 1 + (-0.834 - 1.44i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-0.106 + 0.184i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (0.501 + 0.868i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-1.75 + 3.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 + (2.43 - 4.21i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.97 - 3.41i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.94 + 3.36i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 5.00T + 29T^{2} \)
31 \( 1 + (1.70 - 2.94i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (3.40 + 5.89i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 1.87T + 43T^{2} \)
47 \( 1 + (2.95 + 5.12i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (5.26 - 9.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.39 - 2.41i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.60 + 2.77i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.67 - 6.37i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + (-6.11 + 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.18 - 7.25i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 14.9T + 83T^{2} \)
89 \( 1 + (2.79 + 4.83i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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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

−12.17642571739896604432844253161, −10.92481054361872631810626350024, −10.25760520554516028439285837991, −8.658548369972521111946952141671, −7.81280358275570422685870069404, −7.15524440580226917613682962125, −5.88221694720273527404373553589, −4.87232628467297265312420899679, −4.13477187167716086652833748140, −1.79809989557027757382730496897, 1.83673597647147065510060741841, 3.04324511992030624308654803747, 4.37437500045999051870895406746, 5.06710814444872075159168945406, 7.04665254869646461907436602020, 7.53183043298405100919618596981, 9.292201785514633527060340846226, 9.845980100982525058548354413096, 11.17649505064388613057390394192, 11.72069497862698615189647655796

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