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Properties

Label 2-287-41.10-c1-0-18
Degree $2$
Conductor $287$
Sign $-0.758 - 0.651i$
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  + (−1.51 − 1.10i)2-s − 0.230·3-s + (0.469 + 1.44i)4-s + (−1.06 − 3.26i)5-s + (0.349 + 0.253i)6-s + (0.809 − 0.587i)7-s + (−0.278 + 0.857i)8-s − 2.94·9-s + (−1.99 + 6.12i)10-s + (0.963 − 2.96i)11-s + (−0.108 − 0.332i)12-s + (1.20 + 0.872i)13-s − 1.87·14-s + (0.244 + 0.751i)15-s + (3.82 − 2.78i)16-s + (−0.117 + 0.360i)17-s + ⋯
L(s)  = 1  + (−1.07 − 0.779i)2-s − 0.132·3-s + (0.234 + 0.722i)4-s + (−0.474 − 1.46i)5-s + (0.142 + 0.103i)6-s + (0.305 − 0.222i)7-s + (−0.0985 + 0.303i)8-s − 0.982·9-s + (−0.629 + 1.93i)10-s + (0.290 − 0.893i)11-s + (−0.0312 − 0.0960i)12-s + (0.333 + 0.242i)13-s − 0.501·14-s + (0.0630 + 0.194i)15-s + (0.956 − 0.695i)16-s + (−0.0284 + 0.0874i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.758 - 0.651i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.758 - 0.651i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.103427 + 0.278966i\)
\(L(\frac12)\) \(\approx\) \(0.103427 + 0.278966i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 6.27i)T \)
good2 \( 1 + (1.51 + 1.10i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + 0.230T + 3T^{2} \)
5 \( 1 + (1.06 + 3.26i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-0.963 + 2.96i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-1.20 - 0.872i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.117 - 0.360i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.49 - 4.71i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (5.67 + 4.12i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.49 - 7.68i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.07 - 3.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (0.518 + 1.59i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-0.698 - 0.507i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (0.408 + 0.296i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (3.48 + 10.7i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (0.929 + 0.674i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-4.43 + 3.22i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (4.51 + 13.9i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.61 + 4.98i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 5.18T + 73T^{2} \)
79 \( 1 - 15.0T + 79T^{2} \)
83 \( 1 + 3.04T + 83T^{2} \)
89 \( 1 + (-8.32 + 6.04i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (4.84 + 14.8i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.08461366550154987485266883614, −10.46016214924028985063382781581, −9.097923847868270370175461950544, −8.453744616771053297252788175997, −8.182592373327836728485295412106, −6.17204793344390200595345671778, −5.04385075893149569686729781474, −3.66762032851934325304798016528, −1.76789924231775278707813439396, −0.31384404426978895042122778430, 2.56867855711752102975029902904, 4.08411594725601888989617807267, 6.00269394041671137615790558423, 6.68187021714227623031965766882, 7.65790712104274715667829568892, 8.356412401421438739127862455567, 9.458317373705328090946870710126, 10.40528467035109004542371063276, 11.25028140529342725278471094875, 12.02992059462016243192319506142

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