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Properties

Label 1-927-927.272-r1-0-0
Degree $1$
Conductor $927$
Sign $0.570 - 0.821i$
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  + (−0.213 + 0.976i)2-s + (−0.908 − 0.417i)4-s + (0.0307 + 0.999i)5-s + (−0.389 + 0.920i)7-s + (0.602 − 0.798i)8-s + (−0.982 − 0.183i)10-s + (−0.213 + 0.976i)11-s + (0.992 − 0.122i)13-s + (−0.816 − 0.577i)14-s + (0.650 + 0.759i)16-s + (−0.445 + 0.895i)17-s + (0.932 − 0.361i)19-s + (0.389 − 0.920i)20-s + (−0.908 − 0.417i)22-s + (−0.213 − 0.976i)23-s + ⋯
L(s)  = 1  + (−0.213 + 0.976i)2-s + (−0.908 − 0.417i)4-s + (0.0307 + 0.999i)5-s + (−0.389 + 0.920i)7-s + (0.602 − 0.798i)8-s + (−0.982 − 0.183i)10-s + (−0.213 + 0.976i)11-s + (0.992 − 0.122i)13-s + (−0.816 − 0.577i)14-s + (0.650 + 0.759i)16-s + (−0.445 + 0.895i)17-s + (0.932 − 0.361i)19-s + (0.389 − 0.920i)20-s + (−0.908 − 0.417i)22-s + (−0.213 − 0.976i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.570 - 0.821i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.570 - 0.821i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (272, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.570 - 0.821i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.2850527756 + 0.1491591065i\)
\(L(\frac12)\) \(\approx\) \(-0.2850527756 + 0.1491591065i\)
\(L(1)\) \(\approx\) \(0.4992717923 + 0.5533486123i\)
\(L(1)\) \(\approx\) \(0.4992717923 + 0.5533486123i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.213 + 0.976i)T \)
5 \( 1 + (0.0307 + 0.999i)T \)
7 \( 1 + (-0.389 + 0.920i)T \)
11 \( 1 + (-0.213 + 0.976i)T \)
13 \( 1 + (0.992 - 0.122i)T \)
17 \( 1 + (-0.445 + 0.895i)T \)
19 \( 1 + (0.932 - 0.361i)T \)
23 \( 1 + (-0.213 - 0.976i)T \)
29 \( 1 + (-0.881 - 0.473i)T \)
31 \( 1 + (0.332 - 0.943i)T \)
37 \( 1 + (-0.273 + 0.961i)T \)
41 \( 1 + (-0.881 + 0.473i)T \)
43 \( 1 + (-0.696 + 0.717i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.932 + 0.361i)T \)
59 \( 1 + (0.389 + 0.920i)T \)
61 \( 1 + (-0.998 + 0.0615i)T \)
67 \( 1 + (-0.389 - 0.920i)T \)
71 \( 1 + (0.850 + 0.526i)T \)
73 \( 1 + (-0.850 - 0.526i)T \)
79 \( 1 + (0.881 + 0.473i)T \)
83 \( 1 + (0.389 - 0.920i)T \)
89 \( 1 + (-0.0922 + 0.995i)T \)
97 \( 1 + (0.552 - 0.833i)T \)
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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

−20.74277586742590135315598862042, −20.31888438842440332154952418522, −19.62077035592728118417915537993, −18.74283605047765459472818093725, −17.95596133421215145036967058459, −17.09664717787185914102706093075, −16.30488694832562387574555319414, −15.8318776078310911389867692236, −13.96593777461962427436096869305, −13.67265058887247274971495893205, −13.02631514951547391419488376520, −12.00807554117842547035175903178, −11.28818077079159971852762045137, −10.501866619861866710031244000183, −9.5507849162133450328381442686, −8.8978101768300534740697053968, −8.09791567855338942650192029105, −7.12736819616560838558397038139, −5.67404699896034912021464677127, −4.93878225653828280676552619676, −3.72371055489665248918334359076, −3.35892440527681697837838549330, −1.75775941173709026135325306455, −0.92614799508664967843476531772, −0.08937650636603811778407391794, 1.64025874407695606471203257841, 2.82605527723838251441876401361, 3.91154375693728711308208422351, 4.97892740166705523126954008942, 6.169691842923641993624517217812, 6.38103995805912784260668507695, 7.51975727759801090110215875616, 8.26329570130790585425772417843, 9.26754610356458480546291452749, 9.96490211582837320036279592917, 10.81769608389420912076124129122, 11.890338628045760738833207625734, 12.99148900175898358816490228549, 13.61436973622314625633437281992, 14.712297177629837907345978638980, 15.27743207036725203449907507247, 15.69082086770246156924953619128, 16.77414668779356357685514394318, 17.71753037402346548306736958493, 18.387920407938987978255850337008, 18.741607460445312033756177984119, 19.73757551965330675056719198350, 20.799839063484248528138451778472, 22.00730198843564851894923185146, 22.40958559269524968637756163269

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