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Properties

Label 1-664-664.435-r0-0-0
Degree $1$
Conductor $664$
Sign $0.644 - 0.764i$
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  + (−0.973 + 0.227i)3-s + (0.953 + 0.301i)5-s + (−0.477 + 0.878i)7-s + (0.896 − 0.443i)9-s + (−0.997 + 0.0765i)11-s + (−0.114 − 0.993i)13-s + (−0.997 − 0.0765i)15-s + (0.338 − 0.941i)17-s + (−0.720 − 0.693i)19-s + (0.264 − 0.964i)21-s + (−0.190 − 0.981i)23-s + (0.817 + 0.575i)25-s + (−0.771 + 0.636i)27-s + (−0.0383 − 0.999i)29-s + (0.927 − 0.373i)31-s + ⋯
L(s)  = 1  + (−0.973 + 0.227i)3-s + (0.953 + 0.301i)5-s + (−0.477 + 0.878i)7-s + (0.896 − 0.443i)9-s + (−0.997 + 0.0765i)11-s + (−0.114 − 0.993i)13-s + (−0.997 − 0.0765i)15-s + (0.338 − 0.941i)17-s + (−0.720 − 0.693i)19-s + (0.264 − 0.964i)21-s + (−0.190 − 0.981i)23-s + (0.817 + 0.575i)25-s + (−0.771 + 0.636i)27-s + (−0.0383 − 0.999i)29-s + (0.927 − 0.373i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(664\)    =    \(2^{3} \cdot 83\)
Sign: $0.644 - 0.764i$
Analytic conductor: \(3.08360\)
Root analytic conductor: \(3.08360\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{664} (435, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 664,\ (0:\ ),\ 0.644 - 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7764084458 - 0.3608215847i\)
\(L(\frac12)\) \(\approx\) \(0.7764084458 - 0.3608215847i\)
\(L(1)\) \(\approx\) \(0.8034940300 + 0.01223536030i\)
\(L(1)\) \(\approx\) \(0.8034940300 + 0.01223536030i\)

Euler product

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

−22.97667712596123184134935839917, −22.10993229401929141842540373534, −21.23360242366906607037363580270, −20.83811174237133186016414743875, −19.37922974403623870872996059282, −18.84838527035036062859221412417, −17.66910241659167979799116075920, −17.2715982948386112467522734580, −16.43687662493926730096749761161, −15.86343238250914979857592898500, −14.39628096497611331598057602465, −13.571789980359505952287122778658, −12.87368728403869696307229425265, −12.2078835752459836677572461863, −10.907366463109410146340025639308, −10.32875760571592381005487423016, −9.66100951351958225415968553483, −8.359384007024742296328005138167, −7.22689395529463314218657471152, −6.43364014114678158657206686835, −5.64972500631074622435030268942, −4.760082981368721977711753095127, −3.69529228997843618149285187184, −2.091056385599111970497473308, −1.19012795434376109825881899944, 0.51308014574384370407065133142, 2.27534208533402663057718006101, 2.96320325387543880341276139343, 4.64591771769661012303846967922, 5.43918888399496431336709655499, 6.07488895232060517367447600484, 6.911811603623080495944397095383, 8.165646108260576778158866896476, 9.41877819338103505776144754314, 10.05558901949856954972304456140, 10.757015946163721429116579097317, 11.75013725564678641846772115713, 12.79960352027442579629690065708, 13.14798321069330931181408661995, 14.49266286561861133907693696316, 15.492853489677131097942459782926, 15.96547536213892137047076433596, 17.08156111574657521159127403818, 17.786872774443816873646076381068, 18.39139556830399042641307481881, 19.09597624500201369041476504426, 20.58531732200127740270369905963, 21.21241237655419829175276825723, 21.91730135483787739998719083721, 22.73323481525659701209192103251

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