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Properties

Label 1-927-927.797-r1-0-0
Degree $1$
Conductor $927$
Sign $-0.643 - 0.765i$
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  + (−0.332 + 0.943i)2-s + (−0.779 − 0.626i)4-s + (0.389 − 0.920i)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 + (−0.0307 + 0.999i)13-s + (0.153 + 0.988i)14-s + (0.213 + 0.976i)16-s + (0.273 − 0.961i)17-s + (0.0922 − 0.995i)19-s + (−0.881 + 0.473i)20-s + (−0.779 − 0.626i)22-s + (−0.332 − 0.943i)23-s + ⋯
L(s)  = 1  + (−0.332 + 0.943i)2-s + (−0.779 − 0.626i)4-s + (0.389 − 0.920i)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 + (−0.0307 + 0.999i)13-s + (0.153 + 0.988i)14-s + (0.213 + 0.976i)16-s + (0.273 − 0.961i)17-s + (0.0922 − 0.995i)19-s + (−0.881 + 0.473i)20-s + (−0.779 − 0.626i)22-s + (−0.332 − 0.943i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 927 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.643 - 0.765i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(927\)    =    \(3^{2} \cdot 103\)
Sign: $-0.643 - 0.765i$
Analytic conductor: \(99.6199\)
Root analytic conductor: \(99.6199\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{927} (797, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 927,\ (1:\ ),\ -0.643 - 0.765i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2309698322 - 0.4958928662i\)
\(L(\frac12)\) \(\approx\) \(0.2309698322 - 0.4958928662i\)
\(L(1)\) \(\approx\) \(0.8296904980 + 0.1068371097i\)
\(L(1)\) \(\approx\) \(0.8296904980 + 0.1068371097i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
103 \( 1 \)
good2 \( 1 + (-0.332 + 0.943i)T \)
5 \( 1 + (0.389 - 0.920i)T \)
7 \( 1 + (0.881 - 0.473i)T \)
11 \( 1 + (-0.332 + 0.943i)T \)
13 \( 1 + (-0.0307 + 0.999i)T \)
17 \( 1 + (0.273 - 0.961i)T \)
19 \( 1 + (0.0922 - 0.995i)T \)
23 \( 1 + (-0.332 - 0.943i)T \)
29 \( 1 + (-0.992 + 0.122i)T \)
31 \( 1 + (-0.952 - 0.303i)T \)
37 \( 1 + (0.445 + 0.895i)T \)
41 \( 1 + (-0.992 - 0.122i)T \)
43 \( 1 + (0.552 + 0.833i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (-0.0922 + 0.995i)T \)
59 \( 1 + (-0.881 - 0.473i)T \)
61 \( 1 + (-0.696 - 0.717i)T \)
67 \( 1 + (0.881 + 0.473i)T \)
71 \( 1 + (0.602 - 0.798i)T \)
73 \( 1 + (-0.602 + 0.798i)T \)
79 \( 1 + (0.992 - 0.122i)T \)
83 \( 1 + (-0.881 + 0.473i)T \)
89 \( 1 + (-0.932 - 0.361i)T \)
97 \( 1 + (0.969 + 0.243i)T \)
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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

−21.75707690250997821634526093244, −21.24824225089206853952477108847, −20.47655718988586668142564280335, −19.47323375535976309700495849240, −18.706779709063156797334408774502, −18.20779036292473836346257675269, −17.51614542176859344014897547389, −16.714971013130298543391430453, −15.44364723943645856630904110116, −14.58823512921792692508952867245, −13.92340023063049005930133831170, −13.02022383776645508413551423873, −12.17380384337126514114346456689, −11.16910487725673787048342675044, −10.76087083103158063391000320340, −9.983555423091452891953093100967, −8.97589205353462797522501684371, −8.02341666156486356444657682586, −7.5405243803373027172550904591, −5.803369564586971121871031991817, −5.45571779677964300559385643961, −3.856132481076129826524220560690, −3.22160471041239859424158290917, −2.171327952418621084862681467571, −1.39122291728066000554357115136, 0.13223245581016195475416897925, 1.26248210355078376109581955161, 2.16293911135928689469495828857, 4.17734741008110775500998602165, 4.77119689642860372470697968749, 5.36669767416070811818933420175, 6.62270085292410153070146505926, 7.38963072424632595545365181602, 8.14790577988692637463282495225, 9.14794662826272282737171377850, 9.59309435976459332961618896630, 10.657846558539017144311096669455, 11.69543865486558384752484386989, 12.757611212126216078714473766470, 13.6004924804229893853810874771, 14.21811185298423066358165140204, 15.063100436615494226128278191917, 15.940250972444377632295875485350, 16.84667797555110949327502411460, 17.12547336660065580182747060040, 18.18948149250633084013219433136, 18.56707127579145331313336174106, 20.0929026872712620322775843058, 20.32592590405182029275539200310, 21.39209247755567769234471831807

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