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Properties

Label 1-847-847.139-r0-0-0
Degree $1$
Conductor $847$
Sign $0.641 + 0.767i$
Analytic cond. $3.93345$
Root an. cond. $3.93345$
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  + (−0.774 − 0.633i)2-s + (0.809 + 0.587i)3-s + (0.198 + 0.980i)4-s + (−0.897 − 0.441i)5-s + (−0.254 − 0.967i)6-s + (0.466 − 0.884i)8-s + (0.309 + 0.951i)9-s + (0.415 + 0.909i)10-s + (−0.415 + 0.909i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)15-s + (−0.921 + 0.389i)16-s + (−0.564 + 0.825i)17-s + (0.362 − 0.931i)18-s + (0.941 + 0.336i)19-s + (0.254 − 0.967i)20-s + ⋯
L(s)  = 1  + (−0.774 − 0.633i)2-s + (0.809 + 0.587i)3-s + (0.198 + 0.980i)4-s + (−0.897 − 0.441i)5-s + (−0.254 − 0.967i)6-s + (0.466 − 0.884i)8-s + (0.309 + 0.951i)9-s + (0.415 + 0.909i)10-s + (−0.415 + 0.909i)12-s + (0.993 + 0.113i)13-s + (−0.466 − 0.884i)15-s + (−0.921 + 0.389i)16-s + (−0.564 + 0.825i)17-s + (0.362 − 0.931i)18-s + (0.941 + 0.336i)19-s + (0.254 − 0.967i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.641 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.641 + 0.767i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.641 + 0.767i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9331527396 + 0.4364479165i\)
\(L(\frac12)\) \(\approx\) \(0.9331527396 + 0.4364479165i\)
\(L(1)\) \(\approx\) \(0.8510132555 + 0.04782316130i\)
\(L(1)\) \(\approx\) \(0.8510132555 + 0.04782316130i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.774 - 0.633i)T \)
3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (-0.897 - 0.441i)T \)
13 \( 1 + (0.993 + 0.113i)T \)
17 \( 1 + (-0.564 + 0.825i)T \)
19 \( 1 + (0.941 + 0.336i)T \)
23 \( 1 + (-0.654 - 0.755i)T \)
29 \( 1 + (-0.610 + 0.791i)T \)
31 \( 1 + (0.736 - 0.676i)T \)
37 \( 1 + (-0.870 - 0.491i)T \)
41 \( 1 + (-0.998 - 0.0570i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (0.362 + 0.931i)T \)
53 \( 1 + (-0.921 - 0.389i)T \)
59 \( 1 + (0.998 - 0.0570i)T \)
61 \( 1 + (0.774 - 0.633i)T \)
67 \( 1 + (0.841 - 0.540i)T \)
71 \( 1 + (-0.0285 + 0.999i)T \)
73 \( 1 + (0.974 + 0.226i)T \)
79 \( 1 + (0.985 - 0.170i)T \)
83 \( 1 + (0.516 + 0.856i)T \)
89 \( 1 + (0.959 + 0.281i)T \)
97 \( 1 + (-0.897 + 0.441i)T \)
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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

−22.23859571940084409413270029985, −20.66277111321688605042696514185, −20.238237318400015004080257852354, −19.40902016322902562719817544102, −18.76504130844919581619379221597, −18.16175451157668610463647366248, −17.4670616490392699035211229930, −16.08437277479160065821874116344, −15.60593090418183135256740044801, −15.01322974112436814012136720932, −13.82929014251561439944761690067, −13.599971373757746341783983197575, −11.990154496236613588044317477737, −11.42288460407355214476058313174, −10.329173896536770706388840634323, −9.37369091818970776216846476203, −8.53921916014567035943088956749, −7.91909000337050241257035668160, −7.09848036011124604952106539416, −6.552500732106156672218743799478, −5.30964572780067798353305801748, −3.946220034335621374157792605176, −2.99400218845423213095512415642, −1.8263146775652213644627765620, −0.61658388790092871739415839803, 1.19693605364120475383645493068, 2.28274776374829971458871906340, 3.565515485448517195795217258509, 3.85631770538114308645478186132, 4.96830573461805228295961921460, 6.596442557501624387899721808290, 7.793991462175010501458676582028, 8.27513419341387021090672530525, 8.962515026465465084160038799966, 9.793941909760910447153466364958, 10.7691183755565756418256537578, 11.3461902965996858261801581903, 12.41224583699335598289055402738, 13.14120930262061886752556262547, 14.10856961622707959297951206893, 15.24138142827118873896092651452, 16.00447660409155006128074989552, 16.3919999464835001836949754493, 17.477922446687942020306374864824, 18.585708076194940857359676617677, 19.1094492114622900950284935033, 19.976610751136527723343306694983, 20.483035976718721693879948849920, 21.01115387032941782166447539326, 22.060258655953050551376385256924

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