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Properties

Label 1-927-927.23-r1-0-0
Degree $1$
Conductor $927$
Sign $0.0501 + 0.998i$
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  + (0.389 + 0.920i)2-s + (−0.696 + 0.717i)4-s + (−0.816 + 0.577i)5-s + (−0.153 + 0.988i)7-s + (−0.932 − 0.361i)8-s + (−0.850 − 0.526i)10-s + (0.389 + 0.920i)11-s + (−0.779 − 0.626i)13-s + (−0.969 + 0.243i)14-s + (−0.0307 − 0.999i)16-s + (0.982 − 0.183i)17-s + (0.445 − 0.895i)19-s + (0.153 − 0.988i)20-s + (−0.696 + 0.717i)22-s + (0.389 − 0.920i)23-s + ⋯
L(s)  = 1  + (0.389 + 0.920i)2-s + (−0.696 + 0.717i)4-s + (−0.816 + 0.577i)5-s + (−0.153 + 0.988i)7-s + (−0.932 − 0.361i)8-s + (−0.850 − 0.526i)10-s + (0.389 + 0.920i)11-s + (−0.779 − 0.626i)13-s + (−0.969 + 0.243i)14-s + (−0.0307 − 0.999i)16-s + (0.982 − 0.183i)17-s + (0.445 − 0.895i)19-s + (0.153 − 0.988i)20-s + (−0.696 + 0.717i)22-s + (0.389 − 0.920i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0501 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0501 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $0.0501 + 0.998i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ 0.0501 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.329728973 + 1.264660452i\)
\(L(\frac12)\) \(\approx\) \(1.329728973 + 1.264660452i\)
\(L(1)\) \(\approx\) \(0.8479938979 + 0.6572719322i\)
\(L(1)\) \(\approx\) \(0.8479938979 + 0.6572719322i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (0.389 + 0.920i)T \)
5 \( 1 + (-0.816 + 0.577i)T \)
7 \( 1 + (-0.153 + 0.988i)T \)
11 \( 1 + (0.389 + 0.920i)T \)
13 \( 1 + (-0.779 - 0.626i)T \)
17 \( 1 + (0.982 - 0.183i)T \)
19 \( 1 + (0.445 - 0.895i)T \)
23 \( 1 + (0.389 - 0.920i)T \)
29 \( 1 + (0.908 + 0.417i)T \)
31 \( 1 + (0.881 + 0.473i)T \)
37 \( 1 + (0.739 - 0.673i)T \)
41 \( 1 + (0.908 - 0.417i)T \)
43 \( 1 + (-0.952 + 0.303i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.445 + 0.895i)T \)
59 \( 1 + (0.153 + 0.988i)T \)
61 \( 1 + (0.332 - 0.943i)T \)
67 \( 1 + (-0.153 - 0.988i)T \)
71 \( 1 + (-0.0922 + 0.995i)T \)
73 \( 1 + (0.0922 - 0.995i)T \)
79 \( 1 + (-0.908 - 0.417i)T \)
83 \( 1 + (0.153 - 0.988i)T \)
89 \( 1 + (0.273 - 0.961i)T \)
97 \( 1 + (0.650 - 0.759i)T \)
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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

−21.27248202640729888877252274758, −20.73842539780168526618712307180, −19.84037015942278269094025622820, −19.27305532144758154734568805670, −18.839440032298706908362532151650, −17.43715170054761926838229893328, −16.74495403279688944668526909367, −16.00371130223555963911552738488, −14.82261192189585245566640669038, −14.10625403172165938233037403191, −13.415350370964202917100758208338, −12.495949357144550749809162916184, −11.72453788485145164822298320958, −11.250980564327869115167675807648, −10.0637880182560748398062099618, −9.54852303790347808776213720896, −8.369511543143717598418118716772, −7.650727114207110540227990546246, −6.40823897659648433448561314842, −5.3301634840704023302187695938, −4.38921505812697376096174789635, −3.73662782129482987586263691527, −2.942698985096811501971138776600, −1.31942551375114198387667501494, −0.76570307519689739403619239175, 0.540293000177227508398768891992, 2.61500791082536087521126637482, 3.17335912329480711019278898778, 4.47061050676413413131042078008, 5.08534688773757726687702812071, 6.18896903250340059297573210309, 7.03462630364044051343774207737, 7.66261110144945723307570602143, 8.57120895853333131493485985959, 9.446869003180631595180135435384, 10.391774689444428016308369710322, 11.822831647865540098565483324021, 12.19862730584963433159244138482, 12.944502099453278371498701810460, 14.312899436885933402225174107762, 14.727459357966806239053276473721, 15.47277079343797659948712775445, 15.99558296171039735729552215365, 17.03959905437226812393244220470, 17.90606667555383742413066116015, 18.48963455946439287559420696655, 19.4043449706167521551185645574, 20.21458923798569427786612403986, 21.44767530683114380528684690097, 22.04935832942948678899998668861

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