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Properties

Label 1-287-287.258-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.857 + 0.514i$
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  + (−0.951 − 0.309i)2-s + (−0.707 − 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (0.453 + 0.891i)6-s + (−0.587 − 0.809i)8-s + i·9-s + (0.809 − 0.587i)10-s + (0.987 + 0.156i)11-s + (−0.156 − 0.987i)12-s + (−0.891 + 0.453i)13-s + (0.987 − 0.156i)15-s + (0.309 + 0.951i)16-s + (0.156 − 0.987i)17-s + (0.309 − 0.951i)18-s + (−0.891 − 0.453i)19-s + ⋯
L(s)  = 1  + (−0.951 − 0.309i)2-s + (−0.707 − 0.707i)3-s + (0.809 + 0.587i)4-s + (−0.587 + 0.809i)5-s + (0.453 + 0.891i)6-s + (−0.587 − 0.809i)8-s + i·9-s + (0.809 − 0.587i)10-s + (0.987 + 0.156i)11-s + (−0.156 − 0.987i)12-s + (−0.891 + 0.453i)13-s + (0.987 − 0.156i)15-s + (0.309 + 0.951i)16-s + (0.156 − 0.987i)17-s + (0.309 − 0.951i)18-s + (−0.891 − 0.453i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.857 + 0.514i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.857 + 0.514i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01320518612 + 0.04762435569i\)
\(L(\frac12)\) \(\approx\) \(0.01320518612 + 0.04762435569i\)
\(L(1)\) \(\approx\) \(0.3944905532 - 0.05002633371i\)
\(L(1)\) \(\approx\) \(0.3944905532 - 0.05002633371i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.707 - 0.707i)T \)
5 \( 1 + (-0.587 + 0.809i)T \)
11 \( 1 + (0.987 + 0.156i)T \)
13 \( 1 + (-0.891 + 0.453i)T \)
17 \( 1 + (0.156 - 0.987i)T \)
19 \( 1 + (-0.891 - 0.453i)T \)
23 \( 1 + (-0.309 + 0.951i)T \)
29 \( 1 + (-0.156 - 0.987i)T \)
31 \( 1 + (-0.809 + 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (0.453 + 0.891i)T \)
53 \( 1 + (0.156 + 0.987i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.951 + 0.309i)T \)
67 \( 1 + (-0.987 + 0.156i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 + iT \)
79 \( 1 + (-0.707 - 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (0.453 - 0.891i)T \)
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

−25.18397792823065283743335055050, −24.23531063663506638795345621186, −23.62113944421060318952731089329, −22.49054744840485379050400231964, −21.46292075692729258504093652038, −20.382209019925964587077866128539, −19.76374606862540059252751143929, −18.744353557910018807062570117765, −17.47724497970238008529350019159, −16.803383280478362604051740941940, −16.39645099948782666972436710675, −15.13410580522749388555540317890, −14.68994720314889977509171839726, −12.563584954167717299325338108578, −11.92828788757834679310569288755, −10.85957329445200218711994914053, −10.01524343269579630246971130244, −8.984897569421301896275778283320, −8.23176169734429938206837273673, −6.88533849839207900942757271863, −5.86552334241619985067914602626, −4.79179792467507048245804318368, −3.60982103635311005439137838073, −1.5688189313646799219924639633, −0.04937912902229875504119660387, 1.66212235816131730430292136866, 2.79562581364314104090733565034, 4.27463108249500321299236688705, 6.042355222343040641083689366979, 7.148923222598620176355277605425, 7.39801872732133623446227999329, 8.85244010508425940085928285401, 10.001019230802206335290385777944, 11.04670931443345408693074916180, 11.76522519111166844694740462483, 12.32471442036523257198668706035, 13.8058310015489439609748863027, 15.00814792138266830275306141009, 16.105656884988682618136668934095, 17.06779915250308863331847825568, 17.72586484774175659900244053980, 18.69173396924767365252357852002, 19.36840877213129112794341728178, 19.94269908710686469808239888862, 21.53894620370766353574812487797, 22.24659117040032619815701375986, 23.20464678200582398332045907776, 24.22817210392859544094232227294, 25.07202954987052247847943004565, 25.94892813196703618228448475088

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