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Properties

Label 1-927-927.196-r0-0-0
Degree $1$
Conductor $927$
Sign $-0.831 + 0.555i$
Analytic cond. $4.30496$
Root an. cond. $4.30496$
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  + (0.881 + 0.473i)2-s + (0.552 + 0.833i)4-s + (−0.153 + 0.988i)5-s + (−0.908 + 0.417i)7-s + (0.0922 + 0.995i)8-s + (−0.602 + 0.798i)10-s + (0.881 + 0.473i)11-s + (0.816 + 0.577i)13-s + (−0.998 − 0.0615i)14-s + (−0.389 + 0.920i)16-s + (0.739 − 0.673i)17-s + (−0.273 + 0.961i)19-s + (−0.908 + 0.417i)20-s + (0.552 + 0.833i)22-s + (0.881 − 0.473i)23-s + ⋯
L(s)  = 1  + (0.881 + 0.473i)2-s + (0.552 + 0.833i)4-s + (−0.153 + 0.988i)5-s + (−0.908 + 0.417i)7-s + (0.0922 + 0.995i)8-s + (−0.602 + 0.798i)10-s + (0.881 + 0.473i)11-s + (0.816 + 0.577i)13-s + (−0.998 − 0.0615i)14-s + (−0.389 + 0.920i)16-s + (0.739 − 0.673i)17-s + (−0.273 + 0.961i)19-s + (−0.908 + 0.417i)20-s + (0.552 + 0.833i)22-s + (0.881 − 0.473i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.831 + 0.555i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.831 + 0.555i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (196, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ -0.831 + 0.555i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6684737692 + 2.204687948i\)
\(L(\frac12)\) \(\approx\) \(0.6684737692 + 2.204687948i\)
\(L(1)\) \(\approx\) \(1.270791242 + 1.063728845i\)
\(L(1)\) \(\approx\) \(1.270791242 + 1.063728845i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.881 + 0.473i)T \)
5 \( 1 + (-0.153 + 0.988i)T \)
7 \( 1 + (-0.908 + 0.417i)T \)
11 \( 1 + (0.881 + 0.473i)T \)
13 \( 1 + (0.816 + 0.577i)T \)
17 \( 1 + (0.739 - 0.673i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.881 - 0.473i)T \)
29 \( 1 + (-0.779 - 0.626i)T \)
31 \( 1 + (0.992 - 0.122i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (-0.779 + 0.626i)T \)
43 \( 1 + (0.650 + 0.759i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (-0.273 + 0.961i)T \)
59 \( 1 + (-0.908 - 0.417i)T \)
61 \( 1 + (-0.952 - 0.303i)T \)
67 \( 1 + (-0.908 - 0.417i)T \)
71 \( 1 + (0.932 + 0.361i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (-0.779 - 0.626i)T \)
83 \( 1 + (-0.908 + 0.417i)T \)
89 \( 1 + (0.445 + 0.895i)T \)
97 \( 1 + (0.213 - 0.976i)T \)
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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

−21.42426050738237385011683303479, −20.84941453236985689344099279485, −19.98651699071102612072324565707, −19.44884436639214299695349698932, −18.83886676090145319925857202037, −17.308689623294798198400501111822, −16.71181634878255177347028558000, −15.7998897617169094594505065006, −15.2689180489544222282045752050, −14.0566034267479239547409722436, −13.37235313471159215759593835796, −12.791002010335628252077249542714, −12.07296424568996547974813261119, −11.13671558252752764015203095909, −10.35057537638826106365297636318, −9.34160807977204226051900656726, −8.64984095527794606632925230904, −7.30931258011624676568870356644, −6.35186145162154487791214685251, −5.62522167922287901743671789292, −4.66100801748843954424510664373, −3.64881967587614773961480950515, −3.20801186082253724685955774444, −1.56957687050130871398968674744, −0.77096587237607221875872556349, 1.806247230721516142439017696873, 2.97528703078853415817966482990, 3.56245017254501851924045842449, 4.471528910252634003186105795358, 5.80074379014806780203458795514, 6.435579959497479865198322272544, 7.002671223945511830979694015397, 7.99138005637386939280563795292, 9.10207772355535318857756021107, 10.03354708909845107180988118269, 11.125395399534038372385767760197, 11.89164703299956064174541054483, 12.51726766971297439813033083916, 13.60311956256058739542550466406, 14.20994857143305990009369290050, 15.0148027667199106615505115742, 15.617215298047224837335570623626, 16.53081991986830506766717763444, 17.126465407693130712551643214, 18.42017442631319312075753545825, 18.91551191977395632731089334579, 19.86380561804216360953392253902, 20.91308741255088750640150732373, 21.50490181729154754237418296496, 22.578653008679951916397218009595

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