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Properties

Label 1-847-847.304-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.873 - 0.486i$
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  + (0.710 + 0.703i)2-s + (0.104 − 0.994i)3-s + (0.00951 + 0.999i)4-s + (−0.640 + 0.768i)5-s + (0.774 − 0.633i)6-s + (−0.696 + 0.717i)8-s + (−0.978 − 0.207i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (−0.736 + 0.676i)13-s + (0.696 + 0.717i)15-s + (−0.999 + 0.0190i)16-s + (0.879 − 0.475i)17-s + (−0.548 − 0.836i)18-s + (−0.991 + 0.132i)19-s + (−0.774 − 0.633i)20-s + ⋯
L(s)  = 1  + (0.710 + 0.703i)2-s + (0.104 − 0.994i)3-s + (0.00951 + 0.999i)4-s + (−0.640 + 0.768i)5-s + (0.774 − 0.633i)6-s + (−0.696 + 0.717i)8-s + (−0.978 − 0.207i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (−0.736 + 0.676i)13-s + (0.696 + 0.717i)15-s + (−0.999 + 0.0190i)16-s + (0.879 − 0.475i)17-s + (−0.548 − 0.836i)18-s + (−0.991 + 0.132i)19-s + (−0.774 − 0.633i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.873 - 0.486i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.873 - 0.486i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (304, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.873 - 0.486i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06476022020 + 0.2492421013i\)
\(L(\frac12)\) \(\approx\) \(-0.06476022020 + 0.2492421013i\)
\(L(1)\) \(\approx\) \(0.9096209651 + 0.3216107390i\)
\(L(1)\) \(\approx\) \(0.9096209651 + 0.3216107390i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.710 + 0.703i)T \)
3 \( 1 + (0.104 - 0.994i)T \)
5 \( 1 + (-0.640 + 0.768i)T \)
13 \( 1 + (-0.736 + 0.676i)T \)
17 \( 1 + (0.879 - 0.475i)T \)
19 \( 1 + (-0.991 + 0.132i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.941 + 0.336i)T \)
31 \( 1 + (0.398 + 0.917i)T \)
37 \( 1 + (-0.948 - 0.318i)T \)
41 \( 1 + (-0.362 - 0.931i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (-0.548 + 0.836i)T \)
53 \( 1 + (-0.999 - 0.0190i)T \)
59 \( 1 + (-0.988 + 0.151i)T \)
61 \( 1 + (0.964 + 0.263i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.564 + 0.825i)T \)
73 \( 1 + (-0.905 + 0.424i)T \)
79 \( 1 + (0.0665 - 0.997i)T \)
83 \( 1 + (-0.921 + 0.389i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (0.985 - 0.170i)T \)
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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

−21.515319290853259656080714615029, −20.914761279928062018444916045027, −20.21307133299648394399458590481, −19.55849471038412362601642032269, −18.93763430689912623132466969659, −17.43640102248044592246961193359, −16.737366771217443668500892443504, −15.75366197521856440649349958593, −15.10060650771695074197138764521, −14.583731960778421624190468111493, −13.39015461126257304531681330956, −12.672704288261798724280165486, −11.785086465274595845713720125747, −11.1660198043517986796606512454, −10.10692448838342870981135174554, −9.612184760170258313918091427845, −8.54569009014923584452164757934, −7.66825326861167940328835386215, −6.05566375383965072978158834699, −5.26720653637186308907966833828, −4.536486109330243963394300650621, −3.750209336217406554916632267135, −2.97616382657289973292031649559, −1.66466064504264592518749764466, −0.08242709360918736063782572070, 2.00914565748313259798297569887, 2.91065610879677720287767171089, 3.801097245269683384816794697690, 4.916986347006154185780047557290, 6.01947792665600365848649648227, 6.85111380042475248682098008892, 7.32261045154374037216438373184, 8.15690007478827659344183510665, 8.98847915048155605729031735872, 10.50137722839240982215434180485, 11.55118796582687764226252830824, 12.22603172802950149401897463989, 12.749989220008848989616554337182, 14.10123587500161530731432263223, 14.26346602912005310402114661051, 15.09855702581384266060731753964, 16.10955722785434123691105720142, 16.94705431822444582566799506157, 17.693277719595185582208137575092, 18.725269776327889608820680198863, 19.104987732030664801470920564738, 20.202205890589020707107109444700, 21.073906764791825147989644038124, 22.15964404431806545154178958126, 22.70994995978710364991374056055

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