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Properties

Label 1-847-847.622-r0-0-0
Degree $1$
Conductor $847$
Sign $0.637 - 0.770i$
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  + (−0.941 + 0.336i)2-s + (−0.309 − 0.951i)3-s + (0.774 − 0.633i)4-s + (−0.974 + 0.226i)5-s + (0.610 + 0.791i)6-s + (−0.516 + 0.856i)8-s + (−0.809 + 0.587i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (−0.998 + 0.0570i)13-s + (0.516 + 0.856i)15-s + (0.198 − 0.980i)16-s + (−0.466 + 0.884i)17-s + (0.564 − 0.825i)18-s + (−0.985 + 0.170i)19-s + (−0.610 + 0.791i)20-s + ⋯
L(s)  = 1  + (−0.941 + 0.336i)2-s + (−0.309 − 0.951i)3-s + (0.774 − 0.633i)4-s + (−0.974 + 0.226i)5-s + (0.610 + 0.791i)6-s + (−0.516 + 0.856i)8-s + (−0.809 + 0.587i)9-s + (0.841 − 0.540i)10-s + (−0.841 − 0.540i)12-s + (−0.998 + 0.0570i)13-s + (0.516 + 0.856i)15-s + (0.198 − 0.980i)16-s + (−0.466 + 0.884i)17-s + (0.564 − 0.825i)18-s + (−0.985 + 0.170i)19-s + (−0.610 + 0.791i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.637 - 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.637 - 0.770i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (622, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.637 - 0.770i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3623701282 - 0.1706333703i\)
\(L(\frac12)\) \(\approx\) \(0.3623701282 - 0.1706333703i\)
\(L(1)\) \(\approx\) \(0.4538797647 - 0.05570526481i\)
\(L(1)\) \(\approx\) \(0.4538797647 - 0.05570526481i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.941 + 0.336i)T \)
3 \( 1 + (-0.309 - 0.951i)T \)
5 \( 1 + (-0.974 + 0.226i)T \)
13 \( 1 + (-0.998 + 0.0570i)T \)
17 \( 1 + (-0.466 + 0.884i)T \)
19 \( 1 + (-0.985 + 0.170i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
29 \( 1 + (-0.897 - 0.441i)T \)
31 \( 1 + (0.362 - 0.931i)T \)
37 \( 1 + (-0.254 - 0.967i)T \)
41 \( 1 + (-0.0285 - 0.999i)T \)
43 \( 1 + (0.654 + 0.755i)T \)
47 \( 1 + (0.564 + 0.825i)T \)
53 \( 1 + (0.198 + 0.980i)T \)
59 \( 1 + (0.0285 - 0.999i)T \)
61 \( 1 + (0.941 + 0.336i)T \)
67 \( 1 + (-0.959 - 0.281i)T \)
71 \( 1 + (0.696 - 0.717i)T \)
73 \( 1 + (0.993 - 0.113i)T \)
79 \( 1 + (-0.0855 + 0.996i)T \)
83 \( 1 + (-0.870 + 0.491i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
97 \( 1 + (-0.974 - 0.226i)T \)
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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

−22.136745519001795595008032138231, −21.24563457964419102523180016138, −20.44834596470203988380007624, −19.9280724767247224741116977798, −19.14071952025439696558624857533, −18.2552818742666090908611694531, −17.222569263292832199246887058075, −16.7066137960548017444176908263, −15.96782983378216060206693625114, −15.24274620961444833790269366025, −14.58715718269028641333441317171, −12.939776380009664793845032831402, −12.08054509708178265755299398131, −11.478838473831438169533557170880, −10.68570913552150794183156907888, −9.98190177183965433388626779296, −8.9464991798876368094583564506, −8.50418184719331834303892907318, −7.32448732654926885226769799214, −6.59133400934803199674879634675, −5.0953379487690356209885031173, −4.32617270720767802351050602637, −3.31088441273760887839797257323, −2.408344331834412782726422105802, −0.644108870065421972535409431667, 0.43677820565608617220277012262, 1.81888280185278517739081031987, 2.656083942751003541327352047943, 4.13240102284315539092225063922, 5.47019060105688824809081096700, 6.34799045116919801499693452882, 7.22892566374534454044879470349, 7.73265065156116052184297974491, 8.506605479759391807017534271049, 9.4888244242346872629654779402, 10.781060495676369712883658940616, 11.18069983284336735597228873377, 12.15642484879121586671214626624, 12.79406190318943042797973943247, 14.10830223771119387554464355514, 14.98474379977040813161879531961, 15.55304177344179764218609167167, 16.762638673844641928719831387185, 17.18991951805641137240968337406, 17.98494406370312805524588984226, 19.11014320515215505644917253378, 19.195775671186675270652106724085, 19.909834131719625517454041401713, 20.965418739747501929113099890448, 22.24538513028423446860923938939

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