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Properties

Label 1-847-847.114-r0-0-0
Degree $1$
Conductor $847$
Sign $0.516 + 0.856i$
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  + (0.123 + 0.992i)2-s + (0.913 − 0.406i)3-s + (−0.969 + 0.244i)4-s + (−0.710 + 0.703i)5-s + (0.516 + 0.856i)6-s + (−0.362 − 0.931i)8-s + (0.669 − 0.743i)9-s + (−0.786 − 0.618i)10-s + (−0.786 + 0.618i)12-s + (0.897 − 0.441i)13-s + (−0.362 + 0.931i)15-s + (0.879 − 0.475i)16-s + (0.953 − 0.299i)17-s + (0.820 + 0.572i)18-s + (−0.948 + 0.318i)19-s + (0.516 − 0.856i)20-s + ⋯
L(s)  = 1  + (0.123 + 0.992i)2-s + (0.913 − 0.406i)3-s + (−0.969 + 0.244i)4-s + (−0.710 + 0.703i)5-s + (0.516 + 0.856i)6-s + (−0.362 − 0.931i)8-s + (0.669 − 0.743i)9-s + (−0.786 − 0.618i)10-s + (−0.786 + 0.618i)12-s + (0.897 − 0.441i)13-s + (−0.362 + 0.931i)15-s + (0.879 − 0.475i)16-s + (0.953 − 0.299i)17-s + (0.820 + 0.572i)18-s + (−0.948 + 0.318i)19-s + (0.516 − 0.856i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.516 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.516 + 0.856i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.516 + 0.856i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.565909092 + 0.8845131503i\)
\(L(\frac12)\) \(\approx\) \(1.565909092 + 0.8845131503i\)
\(L(1)\) \(\approx\) \(1.206544874 + 0.5280308020i\)
\(L(1)\) \(\approx\) \(1.206544874 + 0.5280308020i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.123 + 0.992i)T \)
3 \( 1 + (0.913 - 0.406i)T \)
5 \( 1 + (-0.710 + 0.703i)T \)
13 \( 1 + (0.897 - 0.441i)T \)
17 \( 1 + (0.953 - 0.299i)T \)
19 \( 1 + (-0.948 + 0.318i)T \)
23 \( 1 + (0.723 + 0.690i)T \)
29 \( 1 + (-0.870 - 0.491i)T \)
31 \( 1 + (0.345 - 0.938i)T \)
37 \( 1 + (-0.532 + 0.846i)T \)
41 \( 1 + (0.974 - 0.226i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (0.820 - 0.572i)T \)
53 \( 1 + (0.879 + 0.475i)T \)
59 \( 1 + (-0.683 + 0.730i)T \)
61 \( 1 + (0.797 + 0.603i)T \)
67 \( 1 + (-0.327 - 0.945i)T \)
71 \( 1 + (0.993 - 0.113i)T \)
73 \( 1 + (0.380 + 0.924i)T \)
79 \( 1 + (0.161 - 0.986i)T \)
83 \( 1 + (-0.564 + 0.825i)T \)
89 \( 1 + (-0.995 + 0.0950i)T \)
97 \( 1 + (-0.254 + 0.967i)T \)
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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

−21.58619061142371162541965309418, −20.93625038246046551970315586590, −20.59938899798672978055448976608, −19.55342064291749377791869103187, −19.18093204933511749871533828275, −18.43402720730520868781105383942, −17.138243427728201130457780981028, −16.30842081767133629222613490825, −15.419584912039065897289906184409, −14.5500507746349118101024631565, −13.86977848492254047667409201186, −12.7799073490022122873892444054, −12.51134376734008606392362638653, −11.16601653631301745945758650068, −10.66645351553040476143890954020, −9.53826712840576261445884014034, −8.75354770828726338023012430620, −8.38067088859354328464379489513, −7.23117001249944584557340579667, −5.616475477852301983381273017323, −4.599684623253181511621811310160, −3.93467309692572673240598928588, −3.22293981213772010430099935804, −2.053089913430876202895985471006, −1.02058518431966326504661790736, 0.96692223459004997403562781848, 2.64484596517202837791334669202, 3.60998755842569085676994584257, 4.15783765012931785794585258299, 5.65808852822920353698594867838, 6.5024691547277186319389474912, 7.44036329733840359299998823466, 7.89777063707896983601149213357, 8.71139213317174715289851191794, 9.62636289637028574668518751115, 10.648514527570068963760648314176, 11.876231478684916917283766400566, 12.79529244884587012862481303940, 13.56638217656496558393027615464, 14.29055229544153109679019222283, 15.188186494232874120160849612621, 15.38592259400854273072324428638, 16.48511183943502657301205732599, 17.45496506127245956486480331538, 18.51337939942001738872345562734, 18.791938221313955783822743770427, 19.579896663620196778135052368227, 20.76277469865165076722161837547, 21.37774357163648961247174119787, 22.67143526074360701648460346460

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