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Properties

Label 1-287-287.191-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.0872 - 0.996i$
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  + (0.866 − 0.5i)2-s + (0.258 + 0.965i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.707 + 0.707i)6-s − i·8-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)10-s + (−0.258 − 0.965i)11-s + (0.965 + 0.258i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.5 + 0.866i)18-s + (0.258 − 0.965i)19-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.258 + 0.965i)3-s + (0.5 − 0.866i)4-s + (−0.866 + 0.5i)5-s + (0.707 + 0.707i)6-s − i·8-s + (−0.866 + 0.5i)9-s + (−0.5 + 0.866i)10-s + (−0.258 − 0.965i)11-s + (0.965 + 0.258i)12-s + (0.707 − 0.707i)13-s + (−0.707 − 0.707i)15-s + (−0.5 − 0.866i)16-s + (−0.965 + 0.258i)17-s + (−0.5 + 0.866i)18-s + (0.258 − 0.965i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.0872 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0872 - 0.996i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.0872 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457772283 - 1.590962817i\)
\(L(\frac12)\) \(\approx\) \(1.457772283 - 1.590962817i\)
\(L(1)\) \(\approx\) \(1.443347773 - 0.3251901248i\)
\(L(1)\) \(\approx\) \(1.443347773 - 0.3251901248i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
11 \( 1 + (-0.258 - 0.965i)T \)
13 \( 1 + (0.707 - 0.707i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (0.258 - 0.965i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + (0.707 - 0.707i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + iT \)
47 \( 1 + (0.258 - 0.965i)T \)
53 \( 1 + (-0.258 - 0.965i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.866 - 0.5i)T \)
67 \( 1 + (-0.965 + 0.258i)T \)
71 \( 1 + (0.707 - 0.707i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.965 - 0.258i)T \)
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

−25.207683423814224714654019958791, −24.57122413975313896330983024948, −23.60732892714861776110666591101, −23.23143242094206534297642002423, −22.30009665079186335824116577486, −20.69055515083496506647232155962, −20.42238343518210275215629523049, −19.264936291850144039717340778887, −18.21059387513014590760943372812, −17.19852575987816870340038394286, −16.16628507684026829963712041939, −15.34835258977983656168126157828, −14.33636082395452257522412969569, −13.47921181652645358478466600383, −12.46032655016769739889125543422, −12.06627842861837948194099870610, −10.93529924159862048508973135567, −8.89388482787719638588983967313, −8.18743304671759983884302127652, −7.1499603919354934865180622524, −6.49884132544015707723629653734, −5.05056940696340669860686490937, −4.079931573178341953362283912403, −2.84703970927963479815345466301, −1.48984340483029290061165851689, 0.46580388638133058905584440826, 2.62378938987659842092554484405, 3.40044400191536867085105336084, 4.252879070077294723506427900698, 5.35601072326981433133093936893, 6.462258585877800910841259267497, 7.93062994561778970831962261179, 9.050848204913475700812362409017, 10.37043739170849593217869770278, 11.09586978848540665818202545923, 11.60759313121835832027468938993, 13.19711587826647701433265734869, 13.8707655072926856674486922280, 15.13093836569552856497637805605, 15.50559319564164748697717173805, 16.27586691114130105719436328076, 17.8685784556042315238433035015, 19.25132008320568041171095996369, 19.683219073206012637862642385710, 20.701370186918619580017314571099, 21.511058318418764058467896860361, 22.31344256131930415665536745234, 23.01115116250178613471126202434, 23.89867492222646073433330238296, 24.92858242761881866593341169969

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