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Properties

Label 1-66e2-4356.79-r0-0-0
Degree $1$
Conductor $4356$
Sign $0.151 - 0.988i$
Analytic cond. $20.2291$
Root an. cond. $20.2291$
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  + (0.861 − 0.508i)5-s + (0.161 − 0.986i)7-s + (0.991 − 0.132i)13-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (0.888 − 0.458i)23-s + (0.483 − 0.875i)25-s + (0.999 + 0.0190i)29-s + (0.761 − 0.647i)31-s + (−0.362 − 0.931i)35-s + (−0.564 − 0.825i)37-s + (0.830 − 0.556i)41-s + (−0.995 + 0.0950i)43-s + (−0.345 − 0.938i)47-s + (−0.948 − 0.318i)49-s + ⋯
L(s)  = 1  + (0.861 − 0.508i)5-s + (0.161 − 0.986i)7-s + (0.991 − 0.132i)13-s + (−0.0855 + 0.996i)17-s + (−0.921 + 0.389i)19-s + (0.888 − 0.458i)23-s + (0.483 − 0.875i)25-s + (0.999 + 0.0190i)29-s + (0.761 − 0.647i)31-s + (−0.362 − 0.931i)35-s + (−0.564 − 0.825i)37-s + (0.830 − 0.556i)41-s + (−0.995 + 0.0950i)43-s + (−0.345 − 0.938i)47-s + (−0.948 − 0.318i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4356 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.151 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4356\)    =    \(2^{2} \cdot 3^{2} \cdot 11^{2}\)
Sign: $0.151 - 0.988i$
Analytic conductor: \(20.2291\)
Root analytic conductor: \(20.2291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4356} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4356,\ (0:\ ),\ 0.151 - 0.988i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.717537353 - 1.474860058i\)
\(L(\frac12)\) \(\approx\) \(1.717537353 - 1.474860058i\)
\(L(1)\) \(\approx\) \(1.287042826 - 0.3692175046i\)
\(L(1)\) \(\approx\) \(1.287042826 - 0.3692175046i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
11 \( 1 \)
good5 \( 1 + (0.861 - 0.508i)T \)
7 \( 1 + (0.161 - 0.986i)T \)
13 \( 1 + (0.991 - 0.132i)T \)
17 \( 1 + (-0.0855 + 0.996i)T \)
19 \( 1 + (-0.921 + 0.389i)T \)
23 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.999 + 0.0190i)T \)
31 \( 1 + (0.761 - 0.647i)T \)
37 \( 1 + (-0.564 - 0.825i)T \)
41 \( 1 + (0.830 - 0.556i)T \)
43 \( 1 + (-0.995 + 0.0950i)T \)
47 \( 1 + (-0.345 - 0.938i)T \)
53 \( 1 + (-0.998 - 0.0570i)T \)
59 \( 1 + (0.0665 - 0.997i)T \)
61 \( 1 + (-0.272 + 0.962i)T \)
67 \( 1 + (0.786 + 0.618i)T \)
71 \( 1 + (-0.974 - 0.226i)T \)
73 \( 1 + (0.254 - 0.967i)T \)
79 \( 1 + (-0.948 + 0.318i)T \)
83 \( 1 + (0.988 - 0.151i)T \)
89 \( 1 + (-0.654 + 0.755i)T \)
97 \( 1 + (0.861 + 0.508i)T \)
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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

−18.50580182873171150259213627551, −17.76042860818663089932377321811, −17.37942231391418539994731773410, −16.406444886796258486436651133375, −15.62896430542541429564811715779, −15.2084768836375675130877714396, −14.31194178230231234217875537741, −13.811323342541627834988682076895, −13.09890279715352132332354519672, −12.43950832945730014691501284932, −11.44453204929280391624393798783, −11.11823664273146511875133172145, −10.19835079254074839125429375336, −9.527081556461822381766628988695, −8.80929604986140380993704484595, −8.34706401044360905614468145610, −7.2050386841880363132125174679, −6.440274756260563696330412381548, −6.06706679994524351145441471701, −5.08318247776524858794559929915, −4.598616865239876331093401652165, −3.15832187117199671331277507554, −2.85525222485954634235189408352, −1.8968062117724458620535630687, −1.13921075441774763235253782937, 0.65090674310796811179975227685, 1.445760088112943695892813356272, 2.14895301934388431016609508730, 3.26411278302469769156023463987, 4.10056166228485613918537048039, 4.67740920079623675041097148540, 5.600525568016852130254328895002, 6.357396957461053458935330336248, 6.79843571349394016950629270904, 7.9951982148171051088609304762, 8.46269948013327305820376004504, 9.15135136120556015994988472971, 10.15461696506647107469349203932, 10.51016942973891798355925311190, 11.13917556517739579696739049009, 12.191997503523111741944324762865, 12.98883447523851338024756807110, 13.28402968651693557397444535182, 14.07821043232273881268552499639, 14.638602091030398537772563530852, 15.50716847762038704900678924954, 16.3436599785813025779435779666, 16.8783257065347914904971410292, 17.46315407616426486751559067211, 17.92188440580186452195236371493

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