Properties

Label 1-287-287.179-r1-0-0
Degree $1$
Conductor $287$
Sign $0.357 + 0.933i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.357 + 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.357 + 0.933i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.357 + 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2298759996 + 0.1581521797i\)
\(L(\frac12)\) \(\approx\) \(0.2298759996 + 0.1581521797i\)
\(L(1)\) \(\approx\) \(0.4148597609 + 0.001564881453i\)
\(L(1)\) \(\approx\) \(0.4148597609 + 0.001564881453i\)

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

Imaginary part of the first few zeros on the critical line

−25.4786539201595100660213586131, −24.00418725832789933699143121981, −23.17704095100556751622608932998, −22.230540840430630727263223204033, −21.55121512258991037904041326352, −20.7247751371800329055671689392, −19.43964281996137887260160263621, −18.46653303371847090387720705213, −18.233448459282194242566805474079, −16.89615995633834830886241395472, −16.25012406834647269393121404137, −15.28980098815047282778901349497, −13.891167145514739161332172933830, −12.567685993266853222935633208290, −11.70244626502108741882361435345, −11.031967545959500354484998687518, −10.216361781205577951579696980204, −9.37032142759928661852922166543, −7.786486549498290684150949385885, −7.107798472997877318263594233611, −5.83642897533730824950757958196, −4.35186842409869671048295682211, −3.2985532786698667639241778267, −1.94956336838607481154476481619, −0.21064433003560882269179280299, 0.63984537611225382198925721005, 2.02925729257676724266748690276, 4.42321034975339873150665170037, 5.30976356339414448322784938005, 6.12040476540588758716553126900, 7.52802120224445090854951162836, 8.00826029373789463575781297815, 9.34619243352350318332055786683, 10.41772074986594367234483500501, 11.17590794824705979134985323136, 12.580319187905995087828397407188, 13.05589626009217292163281381138, 14.80317877233031552106410520450, 15.67353455289791104564803190807, 16.43321425774905304297455369669, 17.245730870075687257404689419458, 17.91624120358910815696153146942, 18.90123204144367618372281292038, 19.83696027568661006471725809491, 20.761173985652004169621316887476, 22.10986505843720879086963606490, 23.170050892169067376569174092109, 23.88866769558023902105480859525, 24.34859881762918620809375713468, 25.42228813325662096238430825389

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