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Properties

Label 1-1944-1944.931-r1-0-0
Degree $1$
Conductor $1944$
Sign $0.0903 + 0.995i$
Analytic cond. $208.911$
Root an. cond. $208.911$
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  + (−0.987 − 0.154i)5-s + (−0.813 + 0.581i)7-s + (0.856 + 0.516i)11-s + (0.740 + 0.672i)13-s + (−0.286 + 0.957i)17-s + (0.973 + 0.230i)19-s + (−0.0968 + 0.995i)23-s + (0.952 + 0.305i)25-s + (0.790 + 0.612i)29-s + (0.963 + 0.268i)31-s + (0.893 − 0.448i)35-s + (0.835 − 0.549i)37-s + (0.533 − 0.845i)41-s + (−0.981 − 0.192i)43-s + (0.963 − 0.268i)47-s + ⋯
L(s)  = 1  + (−0.987 − 0.154i)5-s + (−0.813 + 0.581i)7-s + (0.856 + 0.516i)11-s + (0.740 + 0.672i)13-s + (−0.286 + 0.957i)17-s + (0.973 + 0.230i)19-s + (−0.0968 + 0.995i)23-s + (0.952 + 0.305i)25-s + (0.790 + 0.612i)29-s + (0.963 + 0.268i)31-s + (0.893 − 0.448i)35-s + (0.835 − 0.549i)37-s + (0.533 − 0.845i)41-s + (−0.981 − 0.192i)43-s + (0.963 − 0.268i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0903 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0903 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $0.0903 + 0.995i$
Analytic conductor: \(208.911\)
Root analytic conductor: \(208.911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (931, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1944,\ (1:\ ),\ 0.0903 + 0.995i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.465609710 + 1.338633078i\)
\(L(\frac12)\) \(\approx\) \(1.465609710 + 1.338633078i\)
\(L(1)\) \(\approx\) \(0.9712166758 + 0.2498227287i\)
\(L(1)\) \(\approx\) \(0.9712166758 + 0.2498227287i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.987 - 0.154i)T \)
7 \( 1 + (-0.813 + 0.581i)T \)
11 \( 1 + (0.856 + 0.516i)T \)
13 \( 1 + (0.740 + 0.672i)T \)
17 \( 1 + (-0.286 + 0.957i)T \)
19 \( 1 + (0.973 + 0.230i)T \)
23 \( 1 + (-0.0968 + 0.995i)T \)
29 \( 1 + (0.790 + 0.612i)T \)
31 \( 1 + (0.963 + 0.268i)T \)
37 \( 1 + (0.835 - 0.549i)T \)
41 \( 1 + (0.533 - 0.845i)T \)
43 \( 1 + (-0.981 - 0.192i)T \)
47 \( 1 + (0.963 - 0.268i)T \)
53 \( 1 + (-0.766 + 0.642i)T \)
59 \( 1 + (0.0193 + 0.999i)T \)
61 \( 1 + (0.565 - 0.824i)T \)
67 \( 1 + (-0.790 + 0.612i)T \)
71 \( 1 + (-0.396 - 0.918i)T \)
73 \( 1 + (0.597 + 0.802i)T \)
79 \( 1 + (0.999 - 0.0387i)T \)
83 \( 1 + (0.533 + 0.845i)T \)
89 \( 1 + (0.396 - 0.918i)T \)
97 \( 1 + (0.987 - 0.154i)T \)
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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

−19.75773227167652877793912184273, −18.989621797522406767468404941560, −18.352698656113043823242147212703, −17.47431089222133679625344377902, −16.467094349381118965596718066628, −16.09632094118292830898312991854, −15.44236198958441166458793262507, −14.50797819214361740493860232023, −13.70519677609698702437093330054, −13.136421330649812663197155397090, −12.09305486274004574432104133334, −11.55375370675776330999258053230, −10.8146448291907809241206181543, −9.9587961303957355055684141335, −9.169678274826260241655793088141, −8.25132436578310643947455289392, −7.62750584233803888118801257642, −6.601651836549752093986381725489, −6.26049330764474470312754548070, −4.87027770243586504022190498804, −4.128788740364103419167216948156, −3.28353856064594410433398094024, −2.765089791674786538907792692876, −0.897969952190078294010668070090, −0.597487770933055909604356883033, 0.8424586267494963999187910511, 1.772493703904580419734297193884, 3.04897580062344387804690264787, 3.76668476827137723617885310022, 4.39817746979923743934055787553, 5.54371799767101389634066048380, 6.41153961134633948415629910947, 7.04858261437694982455210987983, 7.98185256429343871900609015637, 8.856089970503149530199332332800, 9.32906140215317560899372983745, 10.31195596825503107278652344758, 11.26302414024537837181249105608, 12.00234268460447795944190867773, 12.36464687799789159452988845762, 13.34107708725903892705401026333, 14.14770664419056419054766940297, 15.06457482118181826298424178871, 15.68116641699137315873065313468, 16.1932595576660549083970828805, 16.98273579482476293800950363020, 17.89063107471475335406247107375, 18.69175556778929751808052396023, 19.44078828101024584660790534591, 19.75821446948003320521499777102

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