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Properties

Label 1-847-847.580-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.536 - 0.844i$
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  + (0.564 − 0.825i)2-s + (0.809 − 0.587i)3-s + (−0.362 − 0.931i)4-s + (0.921 + 0.389i)5-s + (−0.0285 − 0.999i)6-s + (−0.974 − 0.226i)8-s + (0.309 − 0.951i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (0.774 − 0.633i)13-s + (0.974 − 0.226i)15-s + (−0.736 + 0.676i)16-s + (0.897 − 0.441i)17-s + (−0.610 − 0.791i)18-s + (−0.466 − 0.884i)19-s + (0.0285 − 0.999i)20-s + ⋯
L(s)  = 1  + (0.564 − 0.825i)2-s + (0.809 − 0.587i)3-s + (−0.362 − 0.931i)4-s + (0.921 + 0.389i)5-s + (−0.0285 − 0.999i)6-s + (−0.974 − 0.226i)8-s + (0.309 − 0.951i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (0.774 − 0.633i)13-s + (0.974 − 0.226i)15-s + (−0.736 + 0.676i)16-s + (0.897 − 0.441i)17-s + (−0.610 − 0.791i)18-s + (−0.466 − 0.884i)19-s + (0.0285 − 0.999i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.536 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.536 - 0.844i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (580, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.536 - 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457667332 - 2.653118501i\)
\(L(\frac12)\) \(\approx\) \(1.457667332 - 2.653118501i\)
\(L(1)\) \(\approx\) \(1.535543631 - 1.329166624i\)
\(L(1)\) \(\approx\) \(1.535543631 - 1.329166624i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.564 - 0.825i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
5 \( 1 + (0.921 + 0.389i)T \)
13 \( 1 + (0.774 - 0.633i)T \)
17 \( 1 + (0.897 - 0.441i)T \)
19 \( 1 + (-0.466 - 0.884i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (-0.696 + 0.717i)T \)
31 \( 1 + (0.254 + 0.967i)T \)
37 \( 1 + (-0.998 - 0.0570i)T \)
41 \( 1 + (0.941 - 0.336i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (-0.610 + 0.791i)T \)
53 \( 1 + (-0.736 - 0.676i)T \)
59 \( 1 + (-0.941 - 0.336i)T \)
61 \( 1 + (-0.564 - 0.825i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (-0.985 + 0.170i)T \)
73 \( 1 + (0.198 - 0.980i)T \)
79 \( 1 + (-0.516 - 0.856i)T \)
83 \( 1 + (0.993 + 0.113i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (0.921 - 0.389i)T \)
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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

−22.37939740176207854854551826698, −21.39567545503906065525444954308, −20.99265347689685789928360441986, −20.51063029526654254503192542711, −19.03315895737951408893048947991, −18.43326616906233135361644728060, −17.12740688849971829044065247382, −16.6957722100406798679130566516, −15.964378202652174368294372350975, −14.95875690925203910342206666852, −14.374105037292931150046905184295, −13.66889227990302808326502610832, −13.02179704615915662910796585749, −12.13272761516910383947925237647, −10.75468542246361614259354328788, −9.817915200518661567385604380318, −8.98128708611973135153398631182, −8.3806571941671598950310033169, −7.4745335951993686938023843189, −6.22641888645892501737452076289, −5.63101334630229406170803314737, −4.51039428270089134051553877613, −3.86483558979310912658628890798, −2.76409310000182588042598190983, −1.68666584875902388907935782738, 1.12605480980373286075795053093, 1.86849190633980916903902704091, 3.041964167013550777827375609898, 3.342509303328704475826473512094, 4.85222064042571580078451165118, 5.81936718798833185619870111285, 6.61165032634953627734859273993, 7.65775671587703610683723380922, 8.97181051294570188016752738868, 9.41193077463961143136299415106, 10.46175593406208841418950258724, 11.16016933059754332312594924259, 12.35799392836397850272337126483, 13.01885231857750998126497980112, 13.661320832473673900972872832028, 14.30290981648640534276373928835, 15.001218239958072875591909287458, 15.92783009315095164608408237726, 17.59602401016156278815535658739, 17.94856449304011164190079257305, 18.91262116875764905853452074169, 19.41293363414281393503137911719, 20.42377035293198433680653343581, 21.012697327525117178249390869440, 21.5684089414650315413723250487

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