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Properties

Label 1-1929-1929.1220-r1-0-0
Degree $1$
Conductor $1929$
Sign $0.710 + 0.703i$
Analytic cond. $207.299$
Root an. cond. $207.299$
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  + (−0.722 + 0.691i)2-s + (0.0440 − 0.999i)4-s + (0.848 + 0.529i)5-s + (0.774 − 0.632i)7-s + (0.658 + 0.752i)8-s + (−0.978 + 0.204i)10-s + (0.377 + 0.926i)11-s + (0.508 − 0.861i)13-s + (−0.122 + 0.992i)14-s + (−0.996 − 0.0879i)16-s + (−0.141 + 0.989i)17-s + (0.853 + 0.521i)19-s + (0.566 − 0.824i)20-s + (−0.912 − 0.408i)22-s + (0.598 − 0.801i)23-s + ⋯
L(s)  = 1  + (−0.722 + 0.691i)2-s + (0.0440 − 0.999i)4-s + (0.848 + 0.529i)5-s + (0.774 − 0.632i)7-s + (0.658 + 0.752i)8-s + (−0.978 + 0.204i)10-s + (0.377 + 0.926i)11-s + (0.508 − 0.861i)13-s + (−0.122 + 0.992i)14-s + (−0.996 − 0.0879i)16-s + (−0.141 + 0.989i)17-s + (0.853 + 0.521i)19-s + (0.566 − 0.824i)20-s + (−0.912 − 0.408i)22-s + (0.598 − 0.801i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1929 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1929 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.710 + 0.703i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1929\)    =    \(3 \cdot 643\)
Sign: $0.710 + 0.703i$
Analytic conductor: \(207.299\)
Root analytic conductor: \(207.299\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1929} (1220, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1929,\ (1:\ ),\ 0.710 + 0.703i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.428028123 + 0.9982262119i\)
\(L(\frac12)\) \(\approx\) \(2.428028123 + 0.9982262119i\)
\(L(1)\) \(\approx\) \(1.092365671 + 0.3422735257i\)
\(L(1)\) \(\approx\) \(1.092365671 + 0.3422735257i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
643 \( 1 \)
good2 \( 1 + (-0.722 + 0.691i)T \)
5 \( 1 + (0.848 + 0.529i)T \)
7 \( 1 + (0.774 - 0.632i)T \)
11 \( 1 + (0.377 + 0.926i)T \)
13 \( 1 + (0.508 - 0.861i)T \)
17 \( 1 + (-0.141 + 0.989i)T \)
19 \( 1 + (0.853 + 0.521i)T \)
23 \( 1 + (0.598 - 0.801i)T \)
29 \( 1 + (-0.990 - 0.136i)T \)
31 \( 1 + (0.999 + 0.0391i)T \)
37 \( 1 + (0.999 - 0.0195i)T \)
41 \( 1 + (0.430 - 0.902i)T \)
43 \( 1 + (-0.948 - 0.317i)T \)
47 \( 1 + (0.935 + 0.354i)T \)
53 \( 1 + (0.792 - 0.609i)T \)
59 \( 1 + (-0.900 + 0.435i)T \)
61 \( 1 + (0.967 - 0.251i)T \)
67 \( 1 + (-0.0146 - 0.999i)T \)
71 \( 1 + (0.189 - 0.981i)T \)
73 \( 1 + (-0.265 + 0.964i)T \)
79 \( 1 + (0.945 + 0.326i)T \)
83 \( 1 + (-0.999 + 0.0391i)T \)
89 \( 1 + (0.959 - 0.280i)T \)
97 \( 1 + (-0.549 + 0.835i)T \)
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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

−19.82788012355845632196845123189, −18.83390064108173039212533546147, −18.36670015206790318424931912479, −17.72796919534144452497874471529, −16.95426413599774990828895819832, −16.3714400510043242269812511807, −15.63222847887632078405689484142, −14.42585476236177803159862761479, −13.56604734479108096840660090420, −13.27035731224483232620831619460, −12.01258135643460549167980160350, −11.50088760157278331110145835653, −11.038808999784093963836796456188, −9.808606541818931269578927551521, −9.18642125809157486836305106129, −8.81428376335548246219343778528, −7.9392678085185677364013049063, −6.97605974272922368913051854768, −5.983292652719079438727122748180, −5.11252354236874719412398434684, −4.27718278562269701699949757012, −3.09807434832134659761407742461, −2.34771846298028566090664012856, −1.36924602002949205656898715466, −0.84308352680974744103215861446, 0.800531384782122071429350868667, 1.56558713532707095011528038473, 2.37062416245695557695448077389, 3.750179383340549622890258591732, 4.78332456238629222582239268022, 5.60487215322413873692678863438, 6.32292131220316296482762212280, 7.137816040967627587530065424512, 7.78022862502121798268339286203, 8.58053475729809534411421452756, 9.49220969478042300866122369210, 10.245377479429470750426272430992, 10.65686306427311523436207812229, 11.46623378028210153143864036743, 12.70588570461851470129524616508, 13.58785540614225351113027073614, 14.229281330707038825252157167186, 14.964631945077205050354328747897, 15.32469056501558023302737617130, 16.60815362926931650058813643939, 17.13293562477103236215751658712, 17.7217524690050048092701020434, 18.23663069433761981000633998669, 18.940494046452808838998746101841, 20.008326444507508799025691814174

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