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Properties

Label 1-847-847.703-r0-0-0
Degree $1$
Conductor $847$
Sign $0.309 + 0.950i$
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  + (−0.981 + 0.189i)2-s + (0.5 + 0.866i)3-s + (0.928 − 0.371i)4-s + (0.888 − 0.458i)5-s + (−0.654 − 0.755i)6-s + (−0.841 + 0.540i)8-s + (−0.5 + 0.866i)9-s + (−0.786 + 0.618i)10-s + (0.786 + 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.723 − 0.690i)16-s + (0.580 − 0.814i)17-s + (0.327 − 0.945i)18-s + (0.580 + 0.814i)19-s + (0.654 − 0.755i)20-s + ⋯
L(s)  = 1  + (−0.981 + 0.189i)2-s + (0.5 + 0.866i)3-s + (0.928 − 0.371i)4-s + (0.888 − 0.458i)5-s + (−0.654 − 0.755i)6-s + (−0.841 + 0.540i)8-s + (−0.5 + 0.866i)9-s + (−0.786 + 0.618i)10-s + (0.786 + 0.618i)12-s + (−0.142 + 0.989i)13-s + (0.841 + 0.540i)15-s + (0.723 − 0.690i)16-s + (0.580 − 0.814i)17-s + (0.327 − 0.945i)18-s + (0.580 + 0.814i)19-s + (0.654 − 0.755i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.309 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $0.309 + 0.950i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (703, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ 0.309 + 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.097428252 + 0.7969333974i\)
\(L(\frac12)\) \(\approx\) \(1.097428252 + 0.7969333974i\)
\(L(1)\) \(\approx\) \(0.9347003322 + 0.3548676112i\)
\(L(1)\) \(\approx\) \(0.9347003322 + 0.3548676112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.981 + 0.189i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.888 - 0.458i)T \)
13 \( 1 + (-0.142 + 0.989i)T \)
17 \( 1 + (0.580 - 0.814i)T \)
19 \( 1 + (0.580 + 0.814i)T \)
23 \( 1 + (0.723 - 0.690i)T \)
29 \( 1 + (-0.415 + 0.909i)T \)
31 \( 1 + (0.786 - 0.618i)T \)
37 \( 1 + (0.928 + 0.371i)T \)
41 \( 1 + (-0.654 - 0.755i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (0.327 + 0.945i)T \)
53 \( 1 + (0.723 + 0.690i)T \)
59 \( 1 + (-0.981 - 0.189i)T \)
61 \( 1 + (-0.327 - 0.945i)T \)
67 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (0.415 - 0.909i)T \)
73 \( 1 + (0.235 + 0.971i)T \)
79 \( 1 + (0.888 - 0.458i)T \)
83 \( 1 + (-0.959 + 0.281i)T \)
89 \( 1 + (0.995 + 0.0950i)T \)
97 \( 1 + (-0.841 + 0.540i)T \)
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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

−21.62098240420947226528665032950, −21.04636061796141033765547198356, −20.0498411025628530975428394346, −19.5343405963514800165980177953, −18.63304635655431041727956784405, −18.055548560869552255567896577232, −17.40291083803868038393741862115, −16.79831819893638410939152592563, −15.27675807129652536533813937389, −14.93973129325721448417264021686, −13.642698279768414580036527921764, −13.08864644040912195671125182731, −12.131079329402847246531167092034, −11.24159854662422839999902575884, −10.25549381290579445628725412218, −9.58809207590563356006925627836, −8.70362686172321618490260895262, −7.844306445625653885479785148550, −7.1185933655533470796383786433, −6.280083415163603098810491879059, −5.45339715669033986580409898137, −3.394039758015918613413949691413, −2.778632516695782916752959764579, −1.82394415060737254507344979489, −0.8949209074805345693627987295, 1.22416595444588596286502307637, 2.27286791103173836360437963961, 3.1671680320469012275511971536, 4.61999559304386295491983784514, 5.43711933936227692150300026843, 6.40781466831177587206811131831, 7.51406713438035141218552711830, 8.452098268200167620912924712899, 9.264427206648739572864632172025, 9.67563929195238636469182342953, 10.4546491732774398197053475425, 11.393788429858822596399849081281, 12.34911842770236179337248989291, 13.708889826989741073960060580492, 14.32464902097239077644270507326, 15.11104254272254978371057430998, 16.23186789283426409195085034409, 16.593315791684989279662367346423, 17.22414708993186243854121322506, 18.43741253633950850006300172741, 18.89175793769246210465438918307, 20.09989997026314974044924959424, 20.57857808973204365676675873267, 21.200383558951734306759527495855, 21.95553457117090603636201814761

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