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Properties

Label 1-927-927.382-r1-0-0
Degree $1$
Conductor $927$
Sign $0.628 - 0.777i$
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  + (−0.908 − 0.417i)2-s + (0.650 + 0.759i)4-s + (0.998 − 0.0615i)5-s + (−0.696 − 0.717i)7-s + (−0.273 − 0.961i)8-s + (−0.932 − 0.361i)10-s + (0.908 + 0.417i)11-s + (0.969 − 0.243i)13-s + (0.332 + 0.943i)14-s + (−0.153 + 0.988i)16-s + (−0.602 − 0.798i)17-s + (0.739 − 0.673i)19-s + (0.696 + 0.717i)20-s + (−0.650 − 0.759i)22-s + (−0.908 + 0.417i)23-s + ⋯
L(s)  = 1  + (−0.908 − 0.417i)2-s + (0.650 + 0.759i)4-s + (0.998 − 0.0615i)5-s + (−0.696 − 0.717i)7-s + (−0.273 − 0.961i)8-s + (−0.932 − 0.361i)10-s + (0.908 + 0.417i)11-s + (0.969 − 0.243i)13-s + (0.332 + 0.943i)14-s + (−0.153 + 0.988i)16-s + (−0.602 − 0.798i)17-s + (0.739 − 0.673i)19-s + (0.696 + 0.717i)20-s + (−0.650 − 0.759i)22-s + (−0.908 + 0.417i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.628 - 0.777i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.628 - 0.777i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (382, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.628 - 0.777i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.688123186 - 0.8056563928i\)
\(L(\frac12)\) \(\approx\) \(1.688123186 - 0.8056563928i\)
\(L(1)\) \(\approx\) \(0.9241877769 - 0.2433273595i\)
\(L(1)\) \(\approx\) \(0.9241877769 - 0.2433273595i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.908 - 0.417i)T \)
5 \( 1 + (0.998 - 0.0615i)T \)
7 \( 1 + (-0.696 - 0.717i)T \)
11 \( 1 + (0.908 + 0.417i)T \)
13 \( 1 + (0.969 - 0.243i)T \)
17 \( 1 + (-0.602 - 0.798i)T \)
19 \( 1 + (0.739 - 0.673i)T \)
23 \( 1 + (-0.908 + 0.417i)T \)
29 \( 1 + (0.552 + 0.833i)T \)
31 \( 1 + (0.779 + 0.626i)T \)
37 \( 1 + (0.850 + 0.526i)T \)
41 \( 1 + (0.552 - 0.833i)T \)
43 \( 1 + (0.0307 + 0.999i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.739 + 0.673i)T \)
59 \( 1 + (-0.696 + 0.717i)T \)
61 \( 1 + (0.992 - 0.122i)T \)
67 \( 1 + (0.696 - 0.717i)T \)
71 \( 1 + (-0.445 - 0.895i)T \)
73 \( 1 + (-0.445 - 0.895i)T \)
79 \( 1 + (0.552 + 0.833i)T \)
83 \( 1 + (-0.696 - 0.717i)T \)
89 \( 1 + (0.982 + 0.183i)T \)
97 \( 1 + (-0.389 - 0.920i)T \)
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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

−21.80631641524910370667874776400, −20.88314778115928178715115397751, −20.08315756593562934170317731621, −19.10835818611932451794722404461, −18.63433073021615457054907365878, −17.781270826787931814157523816943, −17.13867157671102294226000355201, −16.22797304926707499699231847901, −15.73880213702768419490518520961, −14.61428695174585930416214142170, −13.98482381295863587667865068325, −13.03627442134441384020767079517, −11.92384543138516381191355759846, −11.11791630811935511525239408000, −10.08182850547881317038123630193, −9.51994765545192261149353098595, −8.76971170876583684288310835944, −8.05914166390705957009261268592, −6.57064487075685900244605762623, −6.20038483222638298291808560451, −5.63808620982555806198546171032, −4.05494556566904949430355716410, −2.72948592898740672025569145409, −1.83586054356762706272863511596, −0.845033271052980749841002913983, 0.74094692773932830309328251391, 1.44840511573135147518227096401, 2.65329409605435900296028592309, 3.50495865808923056857597580488, 4.632671668202739962744279951749, 6.13784305010166560540833630056, 6.68846427045508594210390663999, 7.536910716593056976962706636306, 8.80443206194040366527717863002, 9.37460116479911891256439312099, 10.05273754917879053697666124581, 10.81964173745394352047622248510, 11.71975975369693791666020477398, 12.663508233783846859333165299767, 13.53020143167522946736925125513, 14.05939193740411032731867710376, 15.55041636990763244955078192185, 16.20591797878119253395292188215, 16.95141063821282452745365912484, 17.85520968146465829773155971259, 18.0705863235633569741285435393, 19.27917855496764786681125191243, 20.15582943830055454638786687247, 20.358101040214039237545467717029, 21.490949277583351727201113679976

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