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Properties

Label 1-927-927.97-r0-0-0
Degree $1$
Conductor $927$
Sign $0.505 - 0.862i$
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  + (−0.982 + 0.183i)2-s + (0.932 − 0.361i)4-s + (−0.602 − 0.798i)5-s + (0.881 − 0.473i)7-s + (−0.850 + 0.526i)8-s + (0.739 + 0.673i)10-s + (0.332 − 0.943i)11-s + (0.881 − 0.473i)13-s + (−0.779 + 0.626i)14-s + (0.739 − 0.673i)16-s + (0.969 − 0.243i)17-s + (0.816 + 0.577i)19-s + (−0.850 − 0.526i)20-s + (−0.153 + 0.988i)22-s + (0.650 − 0.759i)23-s + ⋯
L(s)  = 1  + (−0.982 + 0.183i)2-s + (0.932 − 0.361i)4-s + (−0.602 − 0.798i)5-s + (0.881 − 0.473i)7-s + (−0.850 + 0.526i)8-s + (0.739 + 0.673i)10-s + (0.332 − 0.943i)11-s + (0.881 − 0.473i)13-s + (−0.779 + 0.626i)14-s + (0.739 − 0.673i)16-s + (0.969 − 0.243i)17-s + (0.816 + 0.577i)19-s + (−0.850 − 0.526i)20-s + (−0.153 + 0.988i)22-s + (0.650 − 0.759i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.505 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.505 - 0.862i$
Analytic conductor: \(4.30496\)
Root analytic conductor: \(4.30496\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (0:\ ),\ 0.505 - 0.862i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9626297246 - 0.5517205140i\)
\(L(\frac12)\) \(\approx\) \(0.9626297246 - 0.5517205140i\)
\(L(1)\) \(\approx\) \(0.8046694381 - 0.1918226410i\)
\(L(1)\) \(\approx\) \(0.8046694381 - 0.1918226410i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.982 + 0.183i)T \)
5 \( 1 + (-0.602 - 0.798i)T \)
7 \( 1 + (0.881 - 0.473i)T \)
11 \( 1 + (0.332 - 0.943i)T \)
13 \( 1 + (0.881 - 0.473i)T \)
17 \( 1 + (0.969 - 0.243i)T \)
19 \( 1 + (0.816 + 0.577i)T \)
23 \( 1 + (0.650 - 0.759i)T \)
29 \( 1 + (-0.602 - 0.798i)T \)
31 \( 1 + (0.213 + 0.976i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (0.992 + 0.122i)T \)
43 \( 1 + (0.445 - 0.895i)T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.908 + 0.417i)T \)
59 \( 1 + (0.881 + 0.473i)T \)
61 \( 1 + (0.969 - 0.243i)T \)
67 \( 1 + (-0.850 + 0.526i)T \)
71 \( 1 + (0.992 + 0.122i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (-0.389 + 0.920i)T \)
83 \( 1 + (-0.850 - 0.526i)T \)
89 \( 1 + (0.932 + 0.361i)T \)
97 \( 1 + (-0.273 - 0.961i)T \)
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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

−21.78988759198476835231845292538, −21.03947975210760587056076862379, −20.348433361983979623656981695662, −19.4272688452656423018846541083, −18.80653436479101101742040524591, −18.02311767646144119596405113714, −17.61421751734402028167774517206, −16.46690406866453350176286412648, −15.70634759113827689776061475442, −14.904299380496260885073576662971, −14.39426558516804948443508921532, −12.94312260141786130021181852770, −11.89504133644232835162828556490, −11.39661364388315649952931438543, −10.79914820404246069971870353568, −9.676557686024926253274491045383, −9.02189193530110499714916579407, −7.89914446901301539368942131492, −7.46185046877926646729392455348, −6.549991361473574253862128191649, −5.46996483392036867141861790480, −4.09687303707236409697467099751, −3.18151571930695614111287487116, −2.11397997687203562152084129979, −1.181007485236610883084475313735, 0.93249778505047865856828339437, 1.26644486221707451075359912850, 2.970987149025188041949833900642, 3.95535765215937507628997858043, 5.214282705554443166629555933209, 5.939868034013869073387787324295, 7.19395313594721905902030531299, 8.05270159426558062212915480289, 8.399210586274085074306885413244, 9.3335170940346961212140115253, 10.368322675513249861052025825302, 11.24006947814584056329493156234, 11.7046560756415469653421639601, 12.73644312481123893625434954160, 13.92552687078945146367733015749, 14.64873250241456886596125768916, 15.70723007170546558250061072070, 16.32654352264765406272992166240, 16.922650922550624517327638218374, 17.71398232041669824471117372082, 18.67599978995080086800163891006, 19.17875478712183885743937935137, 20.26747760689242677352139344877, 20.691136610232293464963694542132, 21.250748655349150146304002993680

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