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Properties

Label 1-1944-1944.5-r1-0-0
Degree $1$
Conductor $1944$
Sign $-0.782 + 0.622i$
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  + (0.0968 − 0.995i)5-s + (0.925 − 0.378i)7-s + (−0.431 − 0.902i)11-s + (−0.657 − 0.753i)13-s + (−0.396 − 0.918i)17-s + (−0.597 − 0.802i)19-s + (0.135 + 0.990i)23-s + (−0.981 − 0.192i)25-s + (0.0193 − 0.999i)29-s + (0.533 − 0.845i)31-s + (−0.286 − 0.957i)35-s + (0.686 + 0.727i)37-s + (−0.987 − 0.154i)41-s + (0.963 − 0.268i)43-s + (−0.533 − 0.845i)47-s + ⋯
L(s)  = 1  + (0.0968 − 0.995i)5-s + (0.925 − 0.378i)7-s + (−0.431 − 0.902i)11-s + (−0.657 − 0.753i)13-s + (−0.396 − 0.918i)17-s + (−0.597 − 0.802i)19-s + (0.135 + 0.990i)23-s + (−0.981 − 0.192i)25-s + (0.0193 − 0.999i)29-s + (0.533 − 0.845i)31-s + (−0.286 − 0.957i)35-s + (0.686 + 0.727i)37-s + (−0.987 − 0.154i)41-s + (0.963 − 0.268i)43-s + (−0.533 − 0.845i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1944 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.782 + 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1944\)    =    \(2^{3} \cdot 3^{5}\)
Sign: $-0.782 + 0.622i$
Analytic conductor: \(208.911\)
Root analytic conductor: \(208.911\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1944} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1944,\ (1:\ ),\ -0.782 + 0.622i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.4369184742 - 1.250493075i\)
\(L(\frac12)\) \(\approx\) \(-0.4369184742 - 1.250493075i\)
\(L(1)\) \(\approx\) \(0.8782521472 - 0.5143326497i\)
\(L(1)\) \(\approx\) \(0.8782521472 - 0.5143326497i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.0968 - 0.995i)T \)
7 \( 1 + (0.925 - 0.378i)T \)
11 \( 1 + (-0.431 - 0.902i)T \)
13 \( 1 + (-0.657 - 0.753i)T \)
17 \( 1 + (-0.396 - 0.918i)T \)
19 \( 1 + (-0.597 - 0.802i)T \)
23 \( 1 + (0.135 + 0.990i)T \)
29 \( 1 + (0.0193 - 0.999i)T \)
31 \( 1 + (0.533 - 0.845i)T \)
37 \( 1 + (0.686 + 0.727i)T \)
41 \( 1 + (-0.987 - 0.154i)T \)
43 \( 1 + (0.963 - 0.268i)T \)
47 \( 1 + (-0.533 - 0.845i)T \)
53 \( 1 + (-0.939 - 0.342i)T \)
59 \( 1 + (-0.565 - 0.824i)T \)
61 \( 1 + (0.211 + 0.977i)T \)
67 \( 1 + (-0.0193 - 0.999i)T \)
71 \( 1 + (0.0581 + 0.998i)T \)
73 \( 1 + (-0.835 - 0.549i)T \)
79 \( 1 + (-0.360 + 0.932i)T \)
83 \( 1 + (0.987 - 0.154i)T \)
89 \( 1 + (0.0581 - 0.998i)T \)
97 \( 1 + (0.0968 + 0.995i)T \)
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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

−20.35728302361113077455592326789, −19.359667942550509986421406582771, −18.82500295437132490105450350274, −17.99540056148564251256856531433, −17.59755726639688038277383637699, −16.73695735273000655414461053161, −15.750483683722105607049556040196, −14.77769066506114637394954859619, −14.69912002208230570505618862545, −13.922321808312295434629681293796, −12.665050605395696479940318886758, −12.25991222101295503665906618893, −11.22094949898535507051941688275, −10.64317790601975007454314163672, −10.00131102614105447566900277933, −9.00443810599780746169287926719, −8.15759705479805733804286182174, −7.41701141011006541261043065799, −6.643389016333601927333674476410, −5.900456993568254169983453692638, −4.76781340189138453447488651777, −4.26328131352379431669144144229, −3.00778510319234794467005376803, −2.14894387041382222555180850819, −1.59937152798962072458617123624, 0.26255582951749022559798163166, 0.83250864206404289068659641460, 1.99438233872261853610367209328, 2.88293736549899034793321615203, 4.07898172832721261887660145169, 4.90092231929230717530138902766, 5.334798230182298557989821781285, 6.344062302815388910682530919100, 7.56127653937947643662415252858, 8.01741758031248382684382571088, 8.79541524182277584361974387416, 9.60400638613289416276937515624, 10.431071713395784441075449730355, 11.44251292921779586162364513787, 11.7346323865273751907387157148, 12.981310908850747459897351677045, 13.41247839536982926087317997358, 14.059390087633573229818089887472, 15.20490621980387665644234681943, 15.62359803614890269049188269840, 16.61676903898189608017221838270, 17.28391784565434049056816895792, 17.67697119145073890356311535260, 18.67251224314805125053516349804, 19.51905930559318973006007304391

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