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Properties

Label 1-837-837.266-r1-0-0
Degree $1$
Conductor $837$
Sign $-0.979 + 0.203i$
Analytic cond. $89.9481$
Root an. cond. $89.9481$
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  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (0.719 − 0.694i)11-s + (−0.882 + 0.469i)13-s + (−0.961 + 0.275i)14-s + (−0.615 − 0.788i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (0.241 + 0.970i)20-s + (−0.241 + 0.970i)22-s + (0.719 + 0.694i)23-s + ⋯
L(s)  = 1  + (−0.848 + 0.529i)2-s + (0.438 − 0.898i)4-s + (−0.766 + 0.642i)5-s + (0.961 + 0.275i)7-s + (0.104 + 0.994i)8-s + (0.309 − 0.951i)10-s + (0.719 − 0.694i)11-s + (−0.882 + 0.469i)13-s + (−0.961 + 0.275i)14-s + (−0.615 − 0.788i)16-s + (0.104 + 0.994i)17-s + (0.309 − 0.951i)19-s + (0.241 + 0.970i)20-s + (−0.241 + 0.970i)22-s + (0.719 + 0.694i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.979 + 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $-0.979 + 0.203i$
Analytic conductor: \(89.9481\)
Root analytic conductor: \(89.9481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (266, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (1:\ ),\ -0.979 + 0.203i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07859903162 + 0.7659002885i\)
\(L(\frac12)\) \(\approx\) \(0.07859903162 + 0.7659002885i\)
\(L(1)\) \(\approx\) \(0.6165577332 + 0.2848862901i\)
\(L(1)\) \(\approx\) \(0.6165577332 + 0.2848862901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.848 + 0.529i)T \)
5 \( 1 + (-0.766 + 0.642i)T \)
7 \( 1 + (0.961 + 0.275i)T \)
11 \( 1 + (0.719 - 0.694i)T \)
13 \( 1 + (-0.882 + 0.469i)T \)
17 \( 1 + (0.104 + 0.994i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
23 \( 1 + (0.719 + 0.694i)T \)
29 \( 1 + (-0.848 + 0.529i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (-0.990 - 0.139i)T \)
43 \( 1 + (0.848 - 0.529i)T \)
47 \( 1 + (-0.990 + 0.139i)T \)
53 \( 1 + (0.104 + 0.994i)T \)
59 \( 1 + (0.882 - 0.469i)T \)
61 \( 1 + (0.173 - 0.984i)T \)
67 \( 1 + (0.173 + 0.984i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (-0.104 + 0.994i)T \)
79 \( 1 + (0.438 + 0.898i)T \)
83 \( 1 + (-0.0348 - 0.999i)T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (0.961 + 0.275i)T \)
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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

−21.05647793381917036315261789988, −20.68864027550279000872153033114, −19.98059636236186530356260547762, −19.32416190399443053531003372844, −18.36740020117450146705777893038, −17.58549506171878944120914347900, −16.87028521123275972884364640570, −16.27695685203015165382969884025, −15.15201364798551561690757095857, −14.47905383998394128210532718217, −13.146300042715403702533545811855, −12.224410023157970010503398349449, −11.79414129270455390860310692737, −10.9774047496525380196235471886, −9.94325040404022648843082588860, −9.197384058526379197610640191373, −8.26390052574767422072452802030, −7.57752975310067932072602175490, −6.96732869788573815804740293432, −5.231705866174259433172664283230, −4.4009602710547769694594681301, −3.50554831499728956624627937335, −2.201388724818030058826008049879, −1.218151631906888381313331139120, −0.26718068564099594156014234566, 1.09743415033250626015972293822, 2.16032869736406773883556613029, 3.39990722142154584296904928275, 4.658550825351129380308837034728, 5.577495668624577263832144902863, 6.74157983854023301771665105896, 7.30071921947609271949556765916, 8.24664963842699371292508740068, 8.8741988614117408772386407936, 9.86863872705008800770956353259, 11.06816306700051998656421842986, 11.291816846343144174486058169460, 12.216839087452560408805283053722, 13.791800512409015850988583508652, 14.596123047503899387805047075328, 15.07215887052364496714580509154, 15.8231708056522896534137060386, 16.97756856001365853287063050820, 17.345924775570376184963551657490, 18.39953437926116952371993782824, 19.08898216118088962899974237329, 19.580801277495464446988589922124, 20.46712540501733741055162187242, 21.65339949824440081069768451810, 22.23171958945166582490542919264

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