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Properties

Label 1-287-287.167-r0-0-0
Degree $1$
Conductor $287$
Sign $0.652 - 0.758i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  − i·2-s + (0.707 + 0.707i)3-s − 4-s − i·5-s + (0.707 − 0.707i)6-s + i·8-s + i·9-s − 10-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)15-s + 16-s + (0.707 − 0.707i)17-s + 18-s + (0.707 − 0.707i)19-s + ⋯
L(s)  = 1  − i·2-s + (0.707 + 0.707i)3-s − 4-s − i·5-s + (0.707 − 0.707i)6-s + i·8-s + i·9-s − 10-s + (0.707 + 0.707i)11-s + (−0.707 − 0.707i)12-s + (0.707 + 0.707i)13-s + (0.707 − 0.707i)15-s + 16-s + (0.707 − 0.707i)17-s + 18-s + (0.707 − 0.707i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.758i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.652 - 0.758i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (167, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.652 - 0.758i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.368209078 - 0.6279209252i\)
\(L(\frac12)\) \(\approx\) \(1.368209078 - 0.6279209252i\)
\(L(1)\) \(\approx\) \(1.198765398 - 0.4259710896i\)
\(L(1)\) \(\approx\) \(1.198765398 - 0.4259710896i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 - iT \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 - iT \)
11 \( 1 + (0.707 + 0.707i)T \)
13 \( 1 + (0.707 + 0.707i)T \)
17 \( 1 + (0.707 - 0.707i)T \)
19 \( 1 + (0.707 - 0.707i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.707 - 0.707i)T \)
31 \( 1 + T \)
37 \( 1 + T \)
43 \( 1 - iT \)
47 \( 1 + (0.707 - 0.707i)T \)
53 \( 1 + (0.707 + 0.707i)T \)
59 \( 1 - T \)
61 \( 1 + iT \)
67 \( 1 + (-0.707 + 0.707i)T \)
71 \( 1 + (-0.707 - 0.707i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.707 + 0.707i)T \)
97 \( 1 + (-0.707 + 0.707i)T \)
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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

−25.59188618307252832312743475945, −24.88806028666684403094000967781, −23.999143568929462936662372905888, −23.1310833437089750076737661838, −22.33744666026454048038321786468, −21.310431467326596676994583630476, −19.932303994956969624615138429872, −18.96415497554738776860034641550, −18.37399890821025682464240903194, −17.606066637685191739882699428905, −16.392757484532219408745846027299, −15.32069971749511352249371324263, −14.45314508601531803106108438486, −13.97223648336331208858648910724, −12.98831583079696038090321843443, −11.82301307835439125544355833493, −10.36001766437339342403738576206, −9.3045389135751190141477176730, −8.11729391437334818263999291682, −7.63160063013837353966487598230, −6.32142673380223484172344420203, −5.91928401062067203834244718023, −3.83897053458605363938836306821, −3.15935316400359031242977899972, −1.28998706782066233540883555329, 1.291894744594250163221301046834, 2.47983555450524682076211209796, 3.89337955593729565446729635976, 4.45435109218744651477375993809, 5.57912070648586491329381971176, 7.63122166293401536204014827081, 8.74242469617326919347508608342, 9.39213511834289108078667350317, 10.06837111154444168822299881871, 11.488506997667179043614416525, 12.14705430181518892448671868395, 13.50336892567475175594554368699, 13.92379042769314092578856362355, 15.183730632104248449508130106861, 16.29147110437768598575602475732, 17.13530674362817917620065884385, 18.34960259447904489524213685954, 19.418575287051530689743410320574, 20.21670665470094247202424895940, 20.68088404738941600014927310696, 21.53293895438128691234091269103, 22.39792017134838650041732487990, 23.42162905654474626486293478175, 24.57967081459016728706368754970, 25.570103984699882836156693490687

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