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Properties

Label 1-847-847.298-r0-0-0
Degree $1$
Conductor $847$
Sign $0.740 + 0.672i$
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  + (−0.888 + 0.458i)2-s + (−0.5 − 0.866i)3-s + (0.580 − 0.814i)4-s + (0.928 + 0.371i)5-s + (0.841 + 0.540i)6-s + (−0.142 + 0.989i)8-s + (−0.5 + 0.866i)9-s + (−0.995 + 0.0950i)10-s + (−0.995 − 0.0950i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.723 + 0.690i)17-s + (0.0475 − 0.998i)18-s + (0.723 − 0.690i)19-s + (0.841 − 0.540i)20-s + ⋯
L(s)  = 1  + (−0.888 + 0.458i)2-s + (−0.5 − 0.866i)3-s + (0.580 − 0.814i)4-s + (0.928 + 0.371i)5-s + (0.841 + 0.540i)6-s + (−0.142 + 0.989i)8-s + (−0.5 + 0.866i)9-s + (−0.995 + 0.0950i)10-s + (−0.995 − 0.0950i)12-s + (0.415 + 0.909i)13-s + (−0.142 − 0.989i)15-s + (−0.327 − 0.945i)16-s + (0.723 + 0.690i)17-s + (0.0475 − 0.998i)18-s + (0.723 − 0.690i)19-s + (0.841 − 0.540i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.740 + 0.672i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.740 + 0.672i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (298, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.740 + 0.672i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8625651779 + 0.3331530805i\)
\(L(\frac12)\) \(\approx\) \(0.8625651779 + 0.3331530805i\)
\(L(1)\) \(\approx\) \(0.7447870427 + 0.08011837159i\)
\(L(1)\) \(\approx\) \(0.7447870427 + 0.08011837159i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.888 + 0.458i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.928 + 0.371i)T \)
13 \( 1 + (0.415 + 0.909i)T \)
17 \( 1 + (0.723 + 0.690i)T \)
19 \( 1 + (0.723 - 0.690i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.959 - 0.281i)T \)
31 \( 1 + (-0.995 + 0.0950i)T \)
37 \( 1 + (0.580 + 0.814i)T \)
41 \( 1 + (0.841 + 0.540i)T \)
43 \( 1 + (-0.142 + 0.989i)T \)
47 \( 1 + (0.0475 + 0.998i)T \)
53 \( 1 + (-0.327 + 0.945i)T \)
59 \( 1 + (-0.888 - 0.458i)T \)
61 \( 1 + (0.0475 + 0.998i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.959 - 0.281i)T \)
73 \( 1 + (0.981 + 0.189i)T \)
79 \( 1 + (0.928 + 0.371i)T \)
83 \( 1 + (-0.654 - 0.755i)T \)
89 \( 1 + (0.235 - 0.971i)T \)
97 \( 1 + (-0.142 + 0.989i)T \)
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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

−21.89045981559618702758031016598, −20.98396207739853014176634222873, −20.57249494062607971487641155337, −19.89963019237648247420042916607, −18.50116294903897008599869414368, −17.9999895887360943020965259306, −17.28139878740337926318845869012, −16.4719636754152190377569395122, −16.02960446033609408384617707532, −14.978158537637469154692386340692, −13.88858367757822864077252955748, −12.82608536450511347441077933352, −12.07460095802250851661539031841, −11.1342928324591966528733882783, −10.41943653641863527665071772130, −9.61875633725862892870532118247, −9.23993750201771369350748190680, −8.12960741975833303983669930467, −7.12228831855556125988996319128, −5.74728812393805808729782850794, −5.42412747983991792446395690296, −3.872242205890038706059442013985, −3.1269097065712427019136315915, −1.82448449884483146874476766915, −0.6867925276171343468514248297, 1.14519358947925571028875247898, 1.873233365355167952070182329609, 2.874240409989989211879198125623, 4.78432294327252277244633236628, 5.93305980974459257394371846017, 6.240353525247635531695154672729, 7.19385014021242266737123374972, 7.93130029167643326567634710408, 9.02741169223448489520833447338, 9.743442193115504296593922452887, 10.817110409606455002930958898788, 11.28878630772514075543917319652, 12.40966066648670179606078955836, 13.417092022765742500030663079106, 14.19489314317598137296605270259, 14.86082625397396210010000073386, 16.22781739551455531684349105554, 16.75938066779874074431217651633, 17.471174819721899814103198899036, 18.34408971160711754058708305906, 18.58862362320739543131393742206, 19.502910199867222665523718430405, 20.41940401085715744105993117077, 21.41630995817571368379853985771, 22.3154293052459176800171810658

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