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Properties

Label 1-837-837.547-r0-0-0
Degree $1$
Conductor $837$
Sign $0.372 - 0.928i$
Analytic cond. $3.88701$
Root an. cond. $3.88701$
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  + (−0.241 + 0.970i)2-s + (−0.882 − 0.469i)4-s + (0.173 − 0.984i)5-s + (−0.374 − 0.927i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.374 − 0.927i)11-s + (0.961 + 0.275i)13-s + (0.990 − 0.139i)14-s + (0.559 + 0.829i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.615 + 0.788i)20-s + (0.990 − 0.139i)22-s + (0.990 − 0.139i)23-s + ⋯
L(s)  = 1  + (−0.241 + 0.970i)2-s + (−0.882 − 0.469i)4-s + (0.173 − 0.984i)5-s + (−0.374 − 0.927i)7-s + (0.669 − 0.743i)8-s + (0.913 + 0.406i)10-s + (−0.374 − 0.927i)11-s + (0.961 + 0.275i)13-s + (0.990 − 0.139i)14-s + (0.559 + 0.829i)16-s + (0.309 + 0.951i)17-s + (−0.104 − 0.994i)19-s + (−0.615 + 0.788i)20-s + (0.990 − 0.139i)22-s + (0.990 − 0.139i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(837\)    =    \(3^{3} \cdot 31\)
Sign: $0.372 - 0.928i$
Analytic conductor: \(3.88701\)
Root analytic conductor: \(3.88701\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{837} (547, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 837,\ (0:\ ),\ 0.372 - 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8201203025 - 0.5547163053i\)
\(L(\frac12)\) \(\approx\) \(0.8201203025 - 0.5547163053i\)
\(L(1)\) \(\approx\) \(0.8592922911 + 0.02373137301i\)
\(L(1)\) \(\approx\) \(0.8592922911 + 0.02373137301i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
31 \( 1 \)
good2 \( 1 + (-0.241 + 0.970i)T \)
5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.374 - 0.927i)T \)
11 \( 1 + (-0.374 - 0.927i)T \)
13 \( 1 + (0.961 + 0.275i)T \)
17 \( 1 + (0.309 + 0.951i)T \)
19 \( 1 + (-0.104 - 0.994i)T \)
23 \( 1 + (0.990 - 0.139i)T \)
29 \( 1 + (-0.241 + 0.970i)T \)
37 \( 1 + T \)
41 \( 1 + (0.559 - 0.829i)T \)
43 \( 1 + (-0.719 - 0.694i)T \)
47 \( 1 + (0.438 - 0.898i)T \)
53 \( 1 + (-0.978 - 0.207i)T \)
59 \( 1 + (-0.241 - 0.970i)T \)
61 \( 1 + (0.173 + 0.984i)T \)
67 \( 1 + (-0.939 + 0.342i)T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (0.309 - 0.951i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (-0.719 - 0.694i)T \)
89 \( 1 + (-0.978 + 0.207i)T \)
97 \( 1 + (-0.615 + 0.788i)T \)
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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

−22.33132675368037901435108340327, −21.259891120593464290269067390945, −20.89947496252538888696687713662, −19.86912639187281741011933641768, −18.90888698053948344346427582344, −18.41545759283893504675819050612, −17.9791222134990614419672499875, −16.89588949542015339042899184963, −15.770326506144549027924699193422, −14.96662009561078486519027006320, −14.11077185718659265843769624477, −13.14532120497399316668965474486, −12.4963865848838522118164667922, −11.504024700936408012382393458927, −10.94599173340279854636572815318, −9.7918027118299024268841748795, −9.55324401441309429645440985090, −8.27353461535976922564443310657, −7.48415013192691640540532213572, −6.26786894855238165073542246140, −5.37086701602499372326134956958, −4.176235222115238240955849883525, −3.043989851694832276218355253774, −2.57380329542411331928399554192, −1.42116597998765100142594431829, 0.54490734307161648470204760753, 1.435633221798679682221981843509, 3.404361900449217045139461985647, 4.26922111968704772573236886527, 5.219189891952598709143430678132, 6.05700686741103240649928473967, 6.89102362527458179766250972849, 7.91783082903514027674293912757, 8.70622743923336766529113731458, 9.25878748698032271338815421032, 10.42188611675245479295331408364, 11.06808943035051925401633514113, 12.635915888839607647746582172646, 13.34755451492350274347082621948, 13.71047300273477495924691042703, 14.84289973372422534685261815395, 15.84018485549743456998092228563, 16.44533582000905924258155570959, 16.95862035442216452460235337294, 17.72997100881170226026922234985, 18.77001028183163786835652238437, 19.46069837811793014339966060715, 20.32309454275151708849614480835, 21.26451023433884031800981289712, 22.023943002453704178499250551884

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