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Properties

Label 1-1984-1984.1355-r0-0-0
Degree $1$
Conductor $1984$
Sign $0.774 - 0.631i$
Analytic cond. $9.21365$
Root an. cond. $9.21365$
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  + (0.999 + 0.0261i)3-s + (−0.608 − 0.793i)5-s + (−0.998 + 0.0523i)7-s + (0.998 + 0.0523i)9-s + (0.824 + 0.566i)11-s + (0.878 − 0.477i)13-s + (−0.587 − 0.809i)15-s + (−0.207 − 0.978i)17-s + (0.958 − 0.284i)19-s + (−0.999 + 0.0261i)21-s + (0.453 + 0.891i)23-s + (−0.258 + 0.965i)25-s + (0.996 + 0.0784i)27-s + (−0.972 − 0.233i)29-s + (0.809 + 0.587i)33-s + ⋯
L(s)  = 1  + (0.999 + 0.0261i)3-s + (−0.608 − 0.793i)5-s + (−0.998 + 0.0523i)7-s + (0.998 + 0.0523i)9-s + (0.824 + 0.566i)11-s + (0.878 − 0.477i)13-s + (−0.587 − 0.809i)15-s + (−0.207 − 0.978i)17-s + (0.958 − 0.284i)19-s + (−0.999 + 0.0261i)21-s + (0.453 + 0.891i)23-s + (−0.258 + 0.965i)25-s + (0.996 + 0.0784i)27-s + (−0.972 − 0.233i)29-s + (0.809 + 0.587i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1984 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.774 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1984\)    =    \(2^{6} \cdot 31\)
Sign: $0.774 - 0.631i$
Analytic conductor: \(9.21365\)
Root analytic conductor: \(9.21365\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1984} (1355, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1984,\ (0:\ ),\ 0.774 - 0.631i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.026966465 - 0.7216842182i\)
\(L(\frac12)\) \(\approx\) \(2.026966465 - 0.7216842182i\)
\(L(1)\) \(\approx\) \(1.373309214 - 0.2134776374i\)
\(L(1)\) \(\approx\) \(1.373309214 - 0.2134776374i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.999 + 0.0261i)T \)
5 \( 1 + (-0.608 - 0.793i)T \)
7 \( 1 + (-0.998 + 0.0523i)T \)
11 \( 1 + (0.824 + 0.566i)T \)
13 \( 1 + (0.878 - 0.477i)T \)
17 \( 1 + (-0.207 - 0.978i)T \)
19 \( 1 + (0.958 - 0.284i)T \)
23 \( 1 + (0.453 + 0.891i)T \)
29 \( 1 + (-0.972 - 0.233i)T \)
37 \( 1 + (0.991 - 0.130i)T \)
41 \( 1 + (-0.358 + 0.933i)T \)
43 \( 1 + (-0.477 + 0.878i)T \)
47 \( 1 + (-0.587 - 0.809i)T \)
53 \( 1 + (0.333 + 0.942i)T \)
59 \( 1 + (0.725 - 0.688i)T \)
61 \( 1 + (-0.923 + 0.382i)T \)
67 \( 1 + (0.130 - 0.991i)T \)
71 \( 1 + (0.0523 - 0.998i)T \)
73 \( 1 + (0.838 - 0.544i)T \)
79 \( 1 + (-0.207 - 0.978i)T \)
83 \( 1 + (0.725 + 0.688i)T \)
89 \( 1 + (-0.453 + 0.891i)T \)
97 \( 1 + (-0.309 - 0.951i)T \)
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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

−19.91292631106462658031722500900, −19.30347680750904958775859187276, −18.73946190804542239592975360100, −18.3181236422697445483180738301, −16.95848915195227693496664635762, −16.2079834959803391233383811862, −15.66002723629654336543371416257, −14.80981390919216990558506501042, −14.296244607100789672596459434699, −13.50602248921382973768455017417, −12.85757124909268613702933355293, −11.936853096316864549157208920326, −11.093680791916363109716496399934, −10.330507903083384328652225478475, −9.5110882778207587893100424431, −8.764207056855055642481263229045, −8.14518222938297243296145572253, −7.08508346287949760743424274530, −6.64985317953366563358696020247, −5.817247534331518795128288255740, −4.200383041005504652115184890310, −3.66502029645775263748970135009, −3.18381803639319186919192486531, −2.164943512243989890855635212, −1.016248927180286115246204936074, 0.8140593128007042389205590413, 1.70772811586400017841288455338, 3.042351749801144275020116516469, 3.49646503937932074023245146066, 4.34199015124512912361310538156, 5.214435471240731143932129939831, 6.361751073439555381525379044149, 7.25868724519235880677585210881, 7.82032310415618274925270942031, 8.78875272248632748978525457431, 9.488610584066821988505245755508, 9.67382194385307158419380856774, 11.11088005812323120114080452400, 11.856468420823363983804820889911, 12.69127982798946573446397849383, 13.35647777476084031857552510056, 13.764692653019710041265530435625, 15.11869720328001745753401574637, 15.308845338431884314838753409196, 16.284212220022579514590597474639, 16.60355322223451075195599839671, 17.91056805222805457560575421223, 18.53436059264039323622699815258, 19.4616035657307687554698428688, 19.95910520422216110448759354668

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