Properties

Label 2-287-41.37-c1-0-8
Degree $2$
Conductor $287$
Sign $0.654 - 0.756i$
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  + (−1.06 + 0.773i)2-s − 2.00·3-s + (−0.0828 + 0.255i)4-s + (0.544 − 1.67i)5-s + (2.13 − 1.55i)6-s + (0.809 + 0.587i)7-s + (−0.922 − 2.83i)8-s + 1.02·9-s + (0.716 + 2.20i)10-s + (0.156 + 0.482i)11-s + (0.166 − 0.511i)12-s + (3.10 − 2.25i)13-s − 1.31·14-s + (−1.09 + 3.36i)15-s + (2.74 + 1.99i)16-s + (1.32 + 4.06i)17-s + ⋯
L(s)  = 1  + (−0.752 + 0.546i)2-s − 1.15·3-s + (−0.0414 + 0.127i)4-s + (0.243 − 0.749i)5-s + (0.872 − 0.633i)6-s + (0.305 + 0.222i)7-s + (−0.326 − 1.00i)8-s + 0.341·9-s + (0.226 + 0.697i)10-s + (0.0472 + 0.145i)11-s + (0.0480 − 0.147i)12-s + (0.860 − 0.625i)13-s − 0.351·14-s + (−0.282 + 0.868i)15-s + (0.685 + 0.498i)16-s + (0.320 + 0.986i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.654 - 0.756i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.654 - 0.756i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (78, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.654 - 0.756i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.545219 + 0.249252i\)
\(L(\frac12)\) \(\approx\) \(0.545219 + 0.249252i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.809 - 0.587i)T \)
41 \( 1 + (-0.340 + 6.39i)T \)
good2 \( 1 + (1.06 - 0.773i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + 2.00T + 3T^{2} \)
5 \( 1 + (-0.544 + 1.67i)T + (-4.04 - 2.93i)T^{2} \)
11 \( 1 + (-0.156 - 0.482i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (-3.10 + 2.25i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.32 - 4.06i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-3.22 - 2.34i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.62 - 1.18i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.441 - 1.36i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.88 - 8.87i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.23 + 6.89i)T + (-29.9 - 21.7i)T^{2} \)
43 \( 1 + (1.77 - 1.28i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (-8.01 + 5.82i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-3.73 + 11.5i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.72 - 2.70i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-7.32 - 5.31i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (1.69 - 5.20i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (0.159 + 0.491i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 - 3.21T + 73T^{2} \)
79 \( 1 + 0.915T + 79T^{2} \)
83 \( 1 - 8.21T + 83T^{2} \)
89 \( 1 + (-5.13 - 3.73i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.25 - 10.0i)T + (-78.4 - 57.0i)T^{2} \)
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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

−12.11471551816708275739382078275, −10.87534468320522822219654052632, −10.06050283681153018657492496561, −8.838594217556883568959497552448, −8.280988444881697982768485755205, −7.08252666764698743769661203555, −5.92870188177004224720289434904, −5.25985460625049500825645669196, −3.71097125798602910189168220049, −1.07160712891085227944555513061, 0.924319601326896733655813923475, 2.70979215415214153254153369627, 4.66184003793056661748190474192, 5.80795364185341482179024176216, 6.57746421906244945266468875585, 7.913718127559824415898908276435, 9.167943957146624800191307384399, 10.01359373590905827831864406802, 10.90152079229547670625066955304, 11.38722989892751265548514185112

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