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Properties

Label 1-847-847.398-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.427 - 0.904i$
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  + (−0.0855 − 0.996i)2-s + (−0.309 + 0.951i)3-s + (−0.985 + 0.170i)4-s + (0.998 − 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.254 + 0.967i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)10-s + (0.142 − 0.989i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (0.941 − 0.336i)16-s + (−0.870 − 0.491i)17-s + (−0.516 + 0.856i)18-s + (−0.736 − 0.676i)19-s + (−0.974 + 0.226i)20-s + ⋯
L(s)  = 1  + (−0.0855 − 0.996i)2-s + (−0.309 + 0.951i)3-s + (−0.985 + 0.170i)4-s + (0.998 − 0.0570i)5-s + (0.974 + 0.226i)6-s + (0.254 + 0.967i)8-s + (−0.809 − 0.587i)9-s + (−0.142 − 0.989i)10-s + (0.142 − 0.989i)12-s + (0.696 − 0.717i)13-s + (−0.254 + 0.967i)15-s + (0.941 − 0.336i)16-s + (−0.870 − 0.491i)17-s + (−0.516 + 0.856i)18-s + (−0.736 − 0.676i)19-s + (−0.974 + 0.226i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.427 - 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.427 - 0.904i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (398, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.427 - 0.904i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5062815714 - 0.7991472919i\)
\(L(\frac12)\) \(\approx\) \(0.5062815714 - 0.7991472919i\)
\(L(1)\) \(\approx\) \(0.8098848188 - 0.3275630170i\)
\(L(1)\) \(\approx\) \(0.8098848188 - 0.3275630170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.0855 - 0.996i)T \)
3 \( 1 + (-0.309 + 0.951i)T \)
5 \( 1 + (0.998 - 0.0570i)T \)
13 \( 1 + (0.696 - 0.717i)T \)
17 \( 1 + (-0.870 - 0.491i)T \)
19 \( 1 + (-0.736 - 0.676i)T \)
23 \( 1 + (-0.959 - 0.281i)T \)
29 \( 1 + (-0.993 - 0.113i)T \)
31 \( 1 + (0.466 - 0.884i)T \)
37 \( 1 + (0.897 - 0.441i)T \)
41 \( 1 + (-0.921 + 0.389i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.516 - 0.856i)T \)
53 \( 1 + (0.941 + 0.336i)T \)
59 \( 1 + (0.921 + 0.389i)T \)
61 \( 1 + (0.0855 - 0.996i)T \)
67 \( 1 + (-0.654 - 0.755i)T \)
71 \( 1 + (0.198 + 0.980i)T \)
73 \( 1 + (-0.0285 - 0.999i)T \)
79 \( 1 + (0.362 + 0.931i)T \)
83 \( 1 + (0.610 - 0.791i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (0.998 + 0.0570i)T \)
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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

−22.42674897124833535092624538762, −21.91727528813342546206864539305, −20.90725211154534349376864335540, −19.68162042857030769501464083031, −18.812468147671992413780284029422, −18.1924349832090792017436140568, −17.59428160271721549377970747520, −16.836647421022139532117541751104, −16.239150700053921184064783722397, −14.98941579359460733635351134475, −14.221981804588894692464979386689, −13.45207570448902639430224528205, −13.0596044897920702710051332187, −11.97675333886667081147245216582, −10.81082595916093712983598240147, −9.93170319893057240804339205801, −8.81007000454869784638155211320, −8.30106936653753122282732974567, −7.1247330711697456273245024674, −6.37905384845733408820989959142, −5.95051148592437023601096359868, −4.96250133264390435842543136564, −3.76958423782505745383327425630, −2.13221855712711873295031942937, −1.340384824219674900417476225520, 0.467754992402125765960044013222, 1.978553036000110168704764254690, 2.84091417270407690102461164711, 3.91450429560428159065137673781, 4.75816601033682893251512019987, 5.60810648400600882394382283701, 6.44777716604685769194519100826, 8.196867939725118243455488017750, 8.97376756181227777351957540509, 9.69360451251058083202846611165, 10.37564845816469655961333093784, 11.08055516281541272788407055758, 11.80305863329354224142682152860, 13.0903801088561770766809635163, 13.44263973440562705975194594333, 14.55386282613649023904571627240, 15.31972499366930800979286437622, 16.48550386402430500799221216214, 17.17893939481645804168515492698, 17.972839489071485464990901705635, 18.442302217847668970688116776733, 19.91008597936269137846322483026, 20.31985668681424583155669653568, 21.13218062405948605847968981781, 21.73223296748648379623134949424

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