↑

Properties

Label 1-4024-4024.1053-r0-0-0
Degree $1$
Conductor $4024$
Sign $0.948 + 0.316i$
Analytic cond. $18.6873$
Root an. cond. $18.6873$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.943 − 0.331i)3-s + (0.998 − 0.0625i)5-s + (0.620 + 0.784i)7-s + (0.780 − 0.625i)9-s + (0.871 + 0.490i)11-s + (−0.0937 + 0.995i)13-s + (0.920 − 0.389i)15-s + (0.192 + 0.981i)17-s + (−0.640 − 0.768i)19-s + (0.845 + 0.533i)21-s + (0.713 − 0.700i)23-s + (0.992 − 0.124i)25-s + (0.528 − 0.848i)27-s + (0.0562 − 0.998i)29-s + (0.360 + 0.932i)31-s + ⋯
L(s)  = 1  + (0.943 − 0.331i)3-s + (0.998 − 0.0625i)5-s + (0.620 + 0.784i)7-s + (0.780 − 0.625i)9-s + (0.871 + 0.490i)11-s + (−0.0937 + 0.995i)13-s + (0.920 − 0.389i)15-s + (0.192 + 0.981i)17-s + (−0.640 − 0.768i)19-s + (0.845 + 0.533i)21-s + (0.713 − 0.700i)23-s + (0.992 − 0.124i)25-s + (0.528 − 0.848i)27-s + (0.0562 − 0.998i)29-s + (0.360 + 0.932i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.316i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4024\)    =    \(2^{3} \cdot 503\)
Sign: $0.948 + 0.316i$
Analytic conductor: \(18.6873\)
Root analytic conductor: \(18.6873\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4024} (1053, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4024,\ (0:\ ),\ 0.948 + 0.316i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.878186233 + 0.6304797235i\)
\(L(\frac12)\) \(\approx\) \(3.878186233 + 0.6304797235i\)
\(L(1)\) \(\approx\) \(2.029587132 + 0.08944596723i\)
\(L(1)\) \(\approx\) \(2.029587132 + 0.08944596723i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
503 \( 1 \)
good3 \( 1 + (0.943 - 0.331i)T \)
5 \( 1 + (0.998 - 0.0625i)T \)
7 \( 1 + (0.620 + 0.784i)T \)
11 \( 1 + (0.871 + 0.490i)T \)
13 \( 1 + (-0.0937 + 0.995i)T \)
17 \( 1 + (0.192 + 0.981i)T \)
19 \( 1 + (-0.640 - 0.768i)T \)
23 \( 1 + (0.713 - 0.700i)T \)
29 \( 1 + (0.0562 - 0.998i)T \)
31 \( 1 + (0.360 + 0.932i)T \)
37 \( 1 + (-0.337 + 0.941i)T \)
41 \( 1 + (0.539 + 0.842i)T \)
43 \( 1 + (0.549 - 0.835i)T \)
47 \( 1 + (-0.951 - 0.307i)T \)
53 \( 1 + (-0.640 + 0.768i)T \)
59 \( 1 + (-0.143 - 0.989i)T \)
61 \( 1 + (-0.984 + 0.174i)T \)
67 \( 1 + (-0.920 - 0.389i)T \)
71 \( 1 + (-0.253 + 0.967i)T \)
73 \( 1 + (0.429 - 0.902i)T \)
79 \( 1 + (0.452 - 0.891i)T \)
83 \( 1 + (-0.939 - 0.343i)T \)
89 \( 1 + (0.0437 - 0.999i)T \)
97 \( 1 + (-0.772 + 0.635i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.32828049855908186324109275100, −17.815150070881600902500865135380, −16.985273143034630653953687185428, −16.550312551972763070222073739144, −15.64765502277786939531803522557, −14.72226921726026821284506144284, −14.3606987074643940122556735967, −13.83859481919438804762851854291, −13.16079942324248497473162400814, −12.532773131609006230592518047918, −11.330997054220258300449005061430, −10.70608449117978812151366544774, −10.10517545684810421721099273640, −9.363130339216997799493402459297, −8.869400314226058713289100751928, −7.94731607830987766454402787539, −7.404119375377203330710597338802, −6.56048530327189146931026023644, −5.60173075431587621719480544375, −4.93246786448063033512908967718, −4.06315317907804296056073750608, −3.31275596760632067270776581127, −2.60107603715204599351969499528, −1.621704226639231312380257375128, −1.01887368406287874116697138818, 1.30232692712986136364127721844, 1.78367461474072954174522178645, 2.42059699940720455462544511050, 3.2239164741340446420681309956, 4.46901802293511738947351202569, 4.73154012340891045302509376293, 6.13566518553333371176686217672, 6.45401794714703679548143414647, 7.263235320649082244925606909604, 8.31723515733010169858229895520, 8.82304156972421660065032915490, 9.29583979057556781754471355021, 10.00992420092135915991538900738, 10.88073748856222894539831843825, 11.836959458474568504884301715692, 12.45291936587416311137146825762, 13.07599653100851591021387182193, 13.89053251086789367137636778229, 14.39895306592906117549842562651, 14.94483231443932218404946363919, 15.48109559176421251716392846806, 16.64970361799882267709290523128, 17.32796340626879996771076543456, 17.756286080353890875812154843994, 18.66784177588455220441745495274

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