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Properties

Label 2-287-41.32-c1-0-1
Degree $2$
Conductor $287$
Sign $0.940 - 0.339i$
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  − 2.36i·2-s + (−0.345 + 0.345i)3-s − 3.59·4-s + 3.38i·5-s + (0.816 + 0.816i)6-s + (−0.707 + 0.707i)7-s + 3.76i·8-s + 2.76i·9-s + 8.01·10-s + (−4.52 + 4.52i)11-s + (1.24 − 1.24i)12-s + (2.27 − 2.27i)13-s + (1.67 + 1.67i)14-s + (−1.16 − 1.16i)15-s + 1.72·16-s + (1.60 + 1.60i)17-s + ⋯
L(s)  = 1  − 1.67i·2-s + (−0.199 + 0.199i)3-s − 1.79·4-s + 1.51i·5-s + (0.333 + 0.333i)6-s + (−0.267 + 0.267i)7-s + 1.33i·8-s + 0.920i·9-s + 2.53·10-s + (−1.36 + 1.36i)11-s + (0.358 − 0.358i)12-s + (0.630 − 0.630i)13-s + (0.446 + 0.446i)14-s + (−0.301 − 0.301i)15-s + 0.430·16-s + (0.388 + 0.388i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.940 - 0.339i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.940 - 0.339i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.940 - 0.339i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.795129 + 0.139191i\)
\(L(\frac12)\) \(\approx\) \(0.795129 + 0.139191i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.707 - 0.707i)T \)
41 \( 1 + (-6.37 - 0.648i)T \)
good2 \( 1 + 2.36iT - 2T^{2} \)
3 \( 1 + (0.345 - 0.345i)T - 3iT^{2} \)
5 \( 1 - 3.38iT - 5T^{2} \)
11 \( 1 + (4.52 - 4.52i)T - 11iT^{2} \)
13 \( 1 + (-2.27 + 2.27i)T - 13iT^{2} \)
17 \( 1 + (-1.60 - 1.60i)T + 17iT^{2} \)
19 \( 1 + (1.78 + 1.78i)T + 19iT^{2} \)
23 \( 1 - 2.09T + 23T^{2} \)
29 \( 1 + (-5.04 + 5.04i)T - 29iT^{2} \)
31 \( 1 + 7.03T + 31T^{2} \)
37 \( 1 + 1.49T + 37T^{2} \)
43 \( 1 - 8.39iT - 43T^{2} \)
47 \( 1 + (-3.22 - 3.22i)T + 47iT^{2} \)
53 \( 1 + (-2.69 + 2.69i)T - 53iT^{2} \)
59 \( 1 - 11.1T + 59T^{2} \)
61 \( 1 - 3.84iT - 61T^{2} \)
67 \( 1 + (-8.06 - 8.06i)T + 67iT^{2} \)
71 \( 1 + (6.19 - 6.19i)T - 71iT^{2} \)
73 \( 1 + 0.337iT - 73T^{2} \)
79 \( 1 + (6.02 - 6.02i)T - 79iT^{2} \)
83 \( 1 - 8.24T + 83T^{2} \)
89 \( 1 + (-7.27 + 7.27i)T - 89iT^{2} \)
97 \( 1 + (3.77 + 3.77i)T + 97iT^{2} \)
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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

−11.54276678973436708205377939626, −10.76505116255124491127816696533, −10.38189954838712102212854222667, −9.715234989593319118460546329179, −8.160873994259049248432508424535, −7.09267363396603102510827899667, −5.55559853915204750429066269540, −4.28664041251668831510695099989, −2.90714486660941802132255754702, −2.27314553924892676294355947410, 0.62034109987483078342350312692, 3.79322786927964107575147886495, 5.16309276949590121286179164916, 5.73835529792964483770310013527, 6.77746762319335674260581418282, 7.911344873012164605977651353567, 8.743238461001367799989751776534, 9.193987875567839375094257701215, 10.74080569959196984279532760452, 12.11008290131914037009018037021

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