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Properties

Label 1-287-287.146-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.878 - 0.477i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.809 + 0.587i)2-s + 3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + 9-s + (−0.309 − 0.951i)10-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + 3-s + (0.309 − 0.951i)4-s + (−0.309 + 0.951i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + 9-s + (−0.309 − 0.951i)10-s + (−0.309 − 0.951i)11-s + (0.309 − 0.951i)12-s + (−0.809 + 0.587i)13-s + (−0.309 + 0.951i)15-s + (−0.809 − 0.587i)16-s + (0.309 + 0.951i)17-s + (−0.809 + 0.587i)18-s + (−0.809 − 0.587i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.878 - 0.477i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.878 - 0.477i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.878 - 0.477i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.09973438308 + 0.3922472535i\)
\(L(\frac12)\) \(\approx\) \(-0.09973438308 + 0.3922472535i\)
\(L(1)\) \(\approx\) \(0.6909541615 + 0.3170788063i\)
\(L(1)\) \(\approx\) \(0.6909541615 + 0.3170788063i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T \)
3 \( 1 + T \)
5 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (-0.309 - 0.951i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (-0.809 + 0.587i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (0.809 + 0.587i)T \)
67 \( 1 + (-0.309 + 0.951i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (-0.809 - 0.587i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.10933867481969257766080959201, −24.25504594239311002100282697442, −22.91185078577090813788267189948, −21.6543250008996094129138628501, −20.65224602595416765409971609509, −20.3099807125846935750812923608, −19.515676801206284694577756309473, −18.58487909033845915168613572849, −17.61154761788049168243752107880, −16.56628383902406399316805758187, −15.70279989935528786223232837737, −14.721234921977747178647213072174, −13.294065086134494609425951486108, −12.56531778021471122691825418295, −11.83899039115901394379693267149, −10.128724367626091237368117125118, −9.73150862183857028217478618065, −8.529553929236305896492914941091, −7.93777183231159110585240469060, −7.01148247248236620941900030265, −4.89829544353552743538734431884, −3.901337249582878619452822278505, −2.6110382905802752413838161910, −1.63311607551536447200531750055, −0.130865805994061792661887642213, 1.79683916089539915825325947926, 2.86938587576263695799422934832, 4.18413979003777354600154249071, 5.87766794678587174435701997402, 6.95364523295660363523755951953, 7.763658381596522741927953235386, 8.58586994426758352253270320323, 9.64642418221200541399391969867, 10.52288697901707697512154699016, 11.45920457486775422374049525135, 13.093562930676783522053300134759, 14.31460431705312261951197877602, 14.71172391984199038939576731714, 15.648388430552067769993048038254, 16.55288356378906398989774988538, 17.77518427577319458206713306331, 18.74302214131711118646804420668, 19.30304574220655515707093437631, 19.90421806869451938461415835241, 21.3129919283365917233413257252, 22.06548299985530050851480667903, 23.71735063383149196000697174467, 23.99050890834330384040627032552, 25.20215312735938648267494683413, 26.10139397025023528775718693841

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