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Properties

Label 1-664-664.107-r0-0-0
Degree $1$
Conductor $664$
Sign $0.711 - 0.702i$
Analytic cond. $3.08360$
Root an. cond. $3.08360$
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  + (0.606 − 0.795i)3-s + (0.338 + 0.941i)5-s + (0.409 − 0.912i)7-s + (−0.264 − 0.964i)9-s + (0.953 − 0.301i)11-s + (0.896 − 0.443i)13-s + (0.953 + 0.301i)15-s + (0.190 + 0.981i)17-s + (0.997 − 0.0765i)19-s + (−0.477 − 0.878i)21-s + (−0.720 + 0.693i)23-s + (−0.771 + 0.636i)25-s + (−0.927 − 0.373i)27-s + (−0.988 + 0.152i)29-s + (−0.0383 + 0.999i)31-s + ⋯
L(s)  = 1  + (0.606 − 0.795i)3-s + (0.338 + 0.941i)5-s + (0.409 − 0.912i)7-s + (−0.264 − 0.964i)9-s + (0.953 − 0.301i)11-s + (0.896 − 0.443i)13-s + (0.953 + 0.301i)15-s + (0.190 + 0.981i)17-s + (0.997 − 0.0765i)19-s + (−0.477 − 0.878i)21-s + (−0.720 + 0.693i)23-s + (−0.771 + 0.636i)25-s + (−0.927 − 0.373i)27-s + (−0.988 + 0.152i)29-s + (−0.0383 + 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.711 - 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(664\)    =    \(2^{3} \cdot 83\)
Sign: $0.711 - 0.702i$
Analytic conductor: \(3.08360\)
Root analytic conductor: \(3.08360\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{664} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 664,\ (0:\ ),\ 0.711 - 0.702i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.961500440 - 0.8050581527i\)
\(L(\frac12)\) \(\approx\) \(1.961500440 - 0.8050581527i\)
\(L(1)\) \(\approx\) \(1.477418627 - 0.3564832515i\)
\(L(1)\) \(\approx\) \(1.477418627 - 0.3564832515i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
83 \( 1 \)
good3 \( 1 + (0.606 - 0.795i)T \)
5 \( 1 + (0.338 + 0.941i)T \)
7 \( 1 + (0.409 - 0.912i)T \)
11 \( 1 + (0.953 - 0.301i)T \)
13 \( 1 + (0.896 - 0.443i)T \)
17 \( 1 + (0.190 + 0.981i)T \)
19 \( 1 + (0.997 - 0.0765i)T \)
23 \( 1 + (-0.720 + 0.693i)T \)
29 \( 1 + (-0.988 + 0.152i)T \)
31 \( 1 + (-0.0383 + 0.999i)T \)
37 \( 1 + (0.264 - 0.964i)T \)
41 \( 1 + (-0.859 - 0.511i)T \)
43 \( 1 + (0.927 - 0.373i)T \)
47 \( 1 + (-0.973 - 0.227i)T \)
53 \( 1 + (-0.973 + 0.227i)T \)
59 \( 1 + (0.817 + 0.575i)T \)
61 \( 1 + (0.665 - 0.746i)T \)
67 \( 1 + (0.543 + 0.839i)T \)
71 \( 1 + (-0.409 - 0.912i)T \)
73 \( 1 + (0.771 + 0.636i)T \)
79 \( 1 + (0.477 - 0.878i)T \)
89 \( 1 + (0.114 - 0.993i)T \)
97 \( 1 + (0.114 + 0.993i)T \)
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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

−22.45891694985535655334850733614, −22.15169433782455969446978774840, −20.95262478838049578198591464816, −20.6843733579947194382316153262, −19.924685283071336943789317120446, −18.808640066089754549879699104981, −18.037635732221279456370994443596, −16.88750781132318375413718361924, −16.24613950318164747251394960723, −15.55598203998748182324611467, −14.546364859103103727153089207941, −13.91995310942720408094545873614, −12.99788583406390503565033702965, −11.80911292189917438605977370216, −11.33504551149907842726043438476, −9.70796516923908216574271117735, −9.442068174367842256384978714903, −8.56957689490256324681666298638, −7.85573931388659013175948312690, −6.29787607910724230467029565307, −5.33135218926843059995509224776, −4.57020935014239502856419302945, −3.637198990016126192038098406259, −2.36522055500308890650031089282, −1.406784732874941447740192064692, 1.17266630355536783683943875124, 1.93672373197690777255713097570, 3.51814704335360166452997761479, 3.6626001000130160308902195580, 5.62470384257640448062178380667, 6.45156442184562250372592670862, 7.273212676261292434873677436708, 7.98102109250933159484363472576, 9.00695976201425551171618677943, 10.02048344118472805834581886127, 10.97226022613133572489284703781, 11.69398471474007524515785760005, 12.89289635425322695973871381980, 13.749659995186145036613766841912, 14.21785550586131072150737355795, 14.90521493679638774464085157995, 16.04299480736863870320592080125, 17.33445486769165007763314734077, 17.76544350996700563678599988741, 18.61166582301964980226317649583, 19.45428967578692862452178614663, 20.09785777502496012318311050736, 20.94502644227685376999217340495, 21.91922078406409406050437849713, 22.80388729414859386878538687706

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