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Properties

Label 1-927-927.608-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.237 - 0.971i$
Analytic cond. $99.6199$
Root an. cond. $99.6199$
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  + (0.0307 + 0.999i)2-s + (−0.998 + 0.0615i)4-s + (0.779 + 0.626i)5-s + (0.816 + 0.577i)7-s + (−0.0922 − 0.995i)8-s + (−0.602 + 0.798i)10-s + (0.0307 + 0.999i)11-s + (−0.908 + 0.417i)13-s + (−0.552 + 0.833i)14-s + (0.992 − 0.122i)16-s + (−0.739 + 0.673i)17-s + (−0.273 + 0.961i)19-s + (−0.816 − 0.577i)20-s + (−0.998 + 0.0615i)22-s + (0.0307 − 0.999i)23-s + ⋯
L(s)  = 1  + (0.0307 + 0.999i)2-s + (−0.998 + 0.0615i)4-s + (0.779 + 0.626i)5-s + (0.816 + 0.577i)7-s + (−0.0922 − 0.995i)8-s + (−0.602 + 0.798i)10-s + (0.0307 + 0.999i)11-s + (−0.908 + 0.417i)13-s + (−0.552 + 0.833i)14-s + (0.992 − 0.122i)16-s + (−0.739 + 0.673i)17-s + (−0.273 + 0.961i)19-s + (−0.816 − 0.577i)20-s + (−0.998 + 0.0615i)22-s + (0.0307 − 0.999i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.237 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.237 - 0.971i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (608, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.237 - 0.971i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.7345464047 + 0.9357114930i\)
\(L(\frac12)\) \(\approx\) \(-0.7345464047 + 0.9357114930i\)
\(L(1)\) \(\approx\) \(0.6537234906 + 0.7873283303i\)
\(L(1)\) \(\approx\) \(0.6537234906 + 0.7873283303i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.0307 + 0.999i)T \)
5 \( 1 + (0.779 + 0.626i)T \)
7 \( 1 + (0.816 + 0.577i)T \)
11 \( 1 + (0.0307 + 0.999i)T \)
13 \( 1 + (-0.908 + 0.417i)T \)
17 \( 1 + (-0.739 + 0.673i)T \)
19 \( 1 + (-0.273 + 0.961i)T \)
23 \( 1 + (0.0307 - 0.999i)T \)
29 \( 1 + (0.153 - 0.988i)T \)
31 \( 1 + (-0.389 + 0.920i)T \)
37 \( 1 + (-0.982 - 0.183i)T \)
41 \( 1 + (0.153 + 0.988i)T \)
43 \( 1 + (0.332 - 0.943i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.273 - 0.961i)T \)
59 \( 1 + (-0.816 + 0.577i)T \)
61 \( 1 + (0.213 + 0.976i)T \)
67 \( 1 + (0.816 - 0.577i)T \)
71 \( 1 + (-0.932 - 0.361i)T \)
73 \( 1 + (0.932 + 0.361i)T \)
79 \( 1 + (-0.153 + 0.988i)T \)
83 \( 1 + (-0.816 - 0.577i)T \)
89 \( 1 + (-0.445 - 0.895i)T \)
97 \( 1 + (-0.952 + 0.303i)T \)
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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

−21.05380430894837218881091760482, −20.3317743115825403503311195677, −19.787874935678191926620776326664, −18.85788808229123626503971173728, −17.75537592212323842808299055548, −17.47741001261323256384598624074, −16.67992152782526004219520434980, −15.45255520142658345471598442058, −14.25024387662787322906111899105, −13.79242141770257726985325471596, −13.088111383899570654518565972574, −12.24256951277177093423080755564, −11.19608188698686002050927488497, −10.78023231878624489249461847950, −9.668279678495006222560322800448, −9.04543895053716072343407454301, −8.22890899307469575139954939154, −7.165359404023102256174231821117, −5.70012332370130396317755842567, −5.01846051340027088530273666783, −4.29704311744183114316790923488, −3.03793645697062560865278585036, −2.119802523717100568097221339976, −1.127191138236804281886320891467, −0.25359441298303150473596000205, 1.68006684888690317309572889766, 2.444935733573003375041027412321, 4.01411420657009059738345943340, 4.855265090955058529568910249692, 5.66501404844506245257326844781, 6.58056742985702872708728462571, 7.23029675312469939238290071715, 8.2523792036545217601336799601, 9.03418924896920926925968839434, 9.958190940333824913170794316486, 10.60706219437208440615669993268, 12.008075535421352789958352933680, 12.668724317276815747570152763838, 13.72674055467984119222781407798, 14.61704051738562963886902165413, 14.80061707661306231252661544193, 15.693131272626146182800670180034, 16.96957015127385366800788803769, 17.34836581373028569756191152986, 18.15128598182866064126600366925, 18.68108199356965006515986455764, 19.70256325683274869692738988491, 21.01227305682851730639041387118, 21.54657454832314675321001348606, 22.38520256703250816795690377746

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