Properties

Label 1-287-287.27-r0-0-0
Degree $1$
Conductor $287$
Sign $0.364 - 0.931i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.364 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.364 - 0.931i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.364 - 0.931i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1345525219 + 0.09184424007i\)
\(L(\frac12)\) \(\approx\) \(-0.1345525219 + 0.09184424007i\)
\(L(1)\) \(\approx\) \(0.3301980196 + 0.4283484947i\)
\(L(1)\) \(\approx\) \(0.3301980196 + 0.4283484947i\)

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

−24.43663366168104874665418952488, −23.84896422782605492750425089872, −22.99962254805977754063083318135, −21.94388795840939007354977640640, −21.270128511859463002907371159607, −20.09271822778957199081843998187, −19.467263577968518505119990946008, −18.482663781884749590130211735232, −17.568763416641208943888685809098, −16.92243462052052454991896817737, −15.79301041928123578949344993143, −14.17655996223859897678321753652, −13.12693426989001374801650912867, −12.6555902107032729991879369692, −11.84464536315347208767589670660, −10.76937686432656532336167900483, −9.97215592561463513484518406326, −8.445332346558418745166066721150, −7.97510714710621169475163076319, −6.10644340925641820257543513222, −5.21950326286866553441495919440, −4.25454303983743704650102297979, −2.594231876108195957292395572987, −1.443713379873153790707833172771, −0.12055575090607238709663671511, 2.62839574552881041390116045274, 4.243880328251595586029124218756, 4.867361460076184781131147760046, 6.24853893922954146864299664474, 6.82690151850865803310258867206, 7.95941678208819945878566462860, 9.472045116682168879383247447793, 10.03601514387850589910498108767, 11.161266574345947997665721551466, 12.259078741108148294479318181678, 13.58104749637728482890562103002, 14.588918273027568086704039472194, 15.38284485013459916879572575916, 15.96332386118053182068575855605, 17.16267696782819268661822840233, 17.81564275888201962599126654955, 18.55233733679874674618397568773, 19.79652864715833076257056715699, 21.3645474169172130861839568601, 21.921067572341521450089356235760, 22.84623901451677877566594965045, 23.362503610505379775695045947, 24.34220604021405089052328505028, 25.55589389725908469039792265327, 26.55700913016902753409222704318

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