Properties

Label 1-847-847.437-r0-0-0
Degree $1$
Conductor $847$
Sign $-0.732 + 0.680i$
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  + (−0.161 − 0.986i)2-s + (−0.913 − 0.406i)3-s + (−0.948 + 0.318i)4-s + (0.0665 − 0.997i)5-s + (−0.254 + 0.967i)6-s + (0.466 + 0.884i)8-s + (0.669 + 0.743i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.993 − 0.113i)13-s + (−0.466 + 0.884i)15-s + (0.797 − 0.603i)16-s + (−0.432 + 0.901i)17-s + (0.625 − 0.780i)18-s + (−0.179 + 0.983i)19-s + (0.254 + 0.967i)20-s + ⋯
L(s)  = 1  + (−0.161 − 0.986i)2-s + (−0.913 − 0.406i)3-s + (−0.948 + 0.318i)4-s + (0.0665 − 0.997i)5-s + (−0.254 + 0.967i)6-s + (0.466 + 0.884i)8-s + (0.669 + 0.743i)9-s + (−0.995 + 0.0950i)10-s + (0.995 + 0.0950i)12-s + (0.993 − 0.113i)13-s + (−0.466 + 0.884i)15-s + (0.797 − 0.603i)16-s + (−0.432 + 0.901i)17-s + (0.625 − 0.780i)18-s + (−0.179 + 0.983i)19-s + (0.254 + 0.967i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.732 + 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(847\)    =    \(7 \cdot 11^{2}\)
Sign: $-0.732 + 0.680i$
Analytic conductor: \(3.93345\)
Root analytic conductor: \(3.93345\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{847} (437, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 847,\ (0:\ ),\ -0.732 + 0.680i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1791658920 - 0.4563570532i\)
\(L(\frac12)\) \(\approx\) \(-0.1791658920 - 0.4563570532i\)
\(L(1)\) \(\approx\) \(0.4227655940 - 0.4520655779i\)
\(L(1)\) \(\approx\) \(0.4227655940 - 0.4520655779i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.161 - 0.986i)T \)
3 \( 1 + (-0.913 - 0.406i)T \)
5 \( 1 + (0.0665 - 0.997i)T \)
13 \( 1 + (0.993 - 0.113i)T \)
17 \( 1 + (-0.432 + 0.901i)T \)
19 \( 1 + (-0.179 + 0.983i)T \)
23 \( 1 + (-0.327 - 0.945i)T \)
29 \( 1 + (-0.610 - 0.791i)T \)
31 \( 1 + (0.217 - 0.976i)T \)
37 \( 1 + (0.00951 - 0.999i)T \)
41 \( 1 + (-0.998 + 0.0570i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.625 + 0.780i)T \)
53 \( 1 + (0.797 + 0.603i)T \)
59 \( 1 + (-0.449 - 0.893i)T \)
61 \( 1 + (-0.935 + 0.353i)T \)
67 \( 1 + (0.0475 - 0.998i)T \)
71 \( 1 + (-0.0285 - 0.999i)T \)
73 \( 1 + (-0.683 - 0.730i)T \)
79 \( 1 + (-0.640 + 0.768i)T \)
83 \( 1 + (0.516 - 0.856i)T \)
89 \( 1 + (-0.235 + 0.971i)T \)
97 \( 1 + (-0.897 - 0.441i)T \)
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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

−22.77538203019687460651228082968, −21.94264661945445849130840323395, −21.52497309481095316075748370781, −20.17429828304737321991454252851, −19.020484742120651383468895306150, −18.193579707234903459684692462634, −17.88967018608815312654635236135, −16.9972409505391571534330842063, −16.05504180209816478067012925895, −15.54973035395338689960252058077, −14.84887617175224571116160249179, −13.782811833333628683720842717040, −13.1968473066486571686224817779, −11.80124016984693588975649796089, −11.06269467229550998147633317216, −10.27868312079657950268586411832, −9.45200773997579458625983775286, −8.56674779755902354046597865612, −7.182437150904184773427547394584, −6.81143493901532013760239459469, −5.913620494478166837358102778867, −5.124928111012123694167041057120, −4.1277812811689198943571079632, −3.15582797051152080367447343598, −1.32384453874829340952194600631, 0.29826223692725564731733565090, 1.42940532809983989826584144246, 2.13111714961444055173343744282, 3.87380554693723513832571995344, 4.409334435511112085305614132881, 5.58748368333911997884641401816, 6.16858451645187334806645668038, 7.76529668841296508402986022414, 8.39981432746718757072083867287, 9.343056551095666468462125035419, 10.39689325139433051315819827827, 10.952656992567616597579505807988, 11.98785326383215766546618288229, 12.464599654029468626417258195638, 13.23103928739979439180789176080, 13.825390662227231786994192850324, 15.28636949876096644387139255460, 16.4266062062990163659278356677, 16.94947837336693591049029046461, 17.66969778352099270818146083586, 18.548831786719211724132524746659, 19.09222450415482896774543497657, 20.15646794548231949252805683807, 20.79586549189541878590192635135, 21.505181183042057109638464164624

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