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Properties

Label 1-287-287.65-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.996 - 0.0806i$
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  + (0.743 + 0.669i)2-s + (0.258 + 0.965i)3-s + (0.104 + 0.994i)4-s + (0.406 − 0.913i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.913 − 0.406i)10-s + (−0.358 + 0.933i)11-s + (−0.933 + 0.358i)12-s + (0.891 + 0.453i)13-s + (0.987 + 0.156i)15-s + (−0.978 + 0.207i)16-s + (0.933 + 0.358i)17-s + (−0.978 − 0.207i)18-s + (−0.838 − 0.544i)19-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (0.258 + 0.965i)3-s + (0.104 + 0.994i)4-s + (0.406 − 0.913i)5-s + (−0.453 + 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.913 − 0.406i)10-s + (−0.358 + 0.933i)11-s + (−0.933 + 0.358i)12-s + (0.891 + 0.453i)13-s + (0.987 + 0.156i)15-s + (−0.978 + 0.207i)16-s + (0.933 + 0.358i)17-s + (−0.978 − 0.207i)18-s + (−0.838 − 0.544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.996 - 0.0806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.996 - 0.0806i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.996 - 0.0806i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1069659058 + 2.648131507i\)
\(L(\frac12)\) \(\approx\) \(-0.1069659058 + 2.648131507i\)
\(L(1)\) \(\approx\) \(1.095466001 + 1.224768986i\)
\(L(1)\) \(\approx\) \(1.095466001 + 1.224768986i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.406 - 0.913i)T \)
11 \( 1 + (-0.358 + 0.933i)T \)
13 \( 1 + (0.891 + 0.453i)T \)
17 \( 1 + (0.933 + 0.358i)T \)
19 \( 1 + (-0.838 - 0.544i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (0.998 - 0.0523i)T \)
53 \( 1 + (0.777 + 0.629i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (-0.207 - 0.978i)T \)
67 \( 1 + (0.629 - 0.777i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.544 + 0.838i)T \)
97 \( 1 + (0.987 + 0.156i)T \)
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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

−24.80037648343612454465817984008, −23.63929805943255745662261814772, −23.13658692204911005720150552731, −22.19744526201391392290314965520, −21.1798025884910031099766949667, −20.43391661424980644136212746692, −19.23820923740982535270069995897, −18.6085026782604152265528940958, −18.10828866006626300751000263189, −16.563788248044619883853391111788, −15.14415735824198399800427987582, −14.33963427626420656211455630211, −13.61440163086560008164308810429, −12.90670799194643304142794265892, −11.77568364708693107014263154750, −10.923285534516306831855993810836, −10.03531586646166499332589871652, −8.61098298149273721347111482480, −7.42207876331267788281322698772, −6.04079453419068894731151614697, −5.84701387775117610000187681430, −3.78104659619389634277410359957, −2.89246568810851457551008981085, −1.939918825986671046563883453843, −0.56286357793997310403033622478, 1.96052126381836815590987319167, 3.47733248737223558727212467354, 4.435292527203919564977643703352, 5.23200310948340869671775045723, 6.17280553705115132823578593372, 7.69635048007645103789866867924, 8.669315650685925554315195205589, 9.471539226792544396295530656517, 10.72127361614009774553513527875, 11.99941961412568758158437747065, 12.96378185079018716883837835804, 13.82982801431737959237710026310, 14.7978178176007735521287691847, 15.63051296963775848900834829856, 16.4587250356389267470991382628, 17.07903989462276055188018740112, 18.15234933699081536572243690356, 19.94365122977399920564971713561, 20.58384088092683519471209810199, 21.456307527695432028778299340727, 21.89415404103299505868177729015, 23.3962628840196454529596109708, 23.61679511024975912902745553922, 25.1528689288362131561811260776, 25.61956149864665811647705418408

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