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Properties

Label 1-804-804.419-r0-0-0
Degree $1$
Conductor $804$
Sign $0.787 + 0.616i$
Analytic cond. $3.73376$
Root an. cond. $3.73376$
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  + (0.959 + 0.281i)5-s + (0.327 − 0.945i)7-s + (0.235 + 0.971i)11-s + (−0.888 + 0.458i)13-s + (−0.928 − 0.371i)17-s + (0.327 + 0.945i)19-s + (0.580 + 0.814i)23-s + (0.841 + 0.540i)25-s + (0.5 + 0.866i)29-s + (0.888 + 0.458i)31-s + (0.580 − 0.814i)35-s + (−0.5 + 0.866i)37-s + (0.786 − 0.618i)41-s + (0.142 + 0.989i)43-s + (−0.995 + 0.0950i)47-s + ⋯
L(s)  = 1  + (0.959 + 0.281i)5-s + (0.327 − 0.945i)7-s + (0.235 + 0.971i)11-s + (−0.888 + 0.458i)13-s + (−0.928 − 0.371i)17-s + (0.327 + 0.945i)19-s + (0.580 + 0.814i)23-s + (0.841 + 0.540i)25-s + (0.5 + 0.866i)29-s + (0.888 + 0.458i)31-s + (0.580 − 0.814i)35-s + (−0.5 + 0.866i)37-s + (0.786 − 0.618i)41-s + (0.142 + 0.989i)43-s + (−0.995 + 0.0950i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.787 + 0.616i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.787 + 0.616i$
Analytic conductor: \(3.73376\)
Root analytic conductor: \(3.73376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 804,\ (0:\ ),\ 0.787 + 0.616i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.631065013 + 0.5624602941i\)
\(L(\frac12)\) \(\approx\) \(1.631065013 + 0.5624602941i\)
\(L(1)\) \(\approx\) \(1.262518463 + 0.1507859034i\)
\(L(1)\) \(\approx\) \(1.262518463 + 0.1507859034i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
67 \( 1 \)
good5 \( 1 + (0.959 + 0.281i)T \)
7 \( 1 + (0.327 - 0.945i)T \)
11 \( 1 + (0.235 + 0.971i)T \)
13 \( 1 + (-0.888 + 0.458i)T \)
17 \( 1 + (-0.928 - 0.371i)T \)
19 \( 1 + (0.327 + 0.945i)T \)
23 \( 1 + (0.580 + 0.814i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + (0.888 + 0.458i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.786 - 0.618i)T \)
43 \( 1 + (0.142 + 0.989i)T \)
47 \( 1 + (-0.995 + 0.0950i)T \)
53 \( 1 + (0.142 - 0.989i)T \)
59 \( 1 + (0.841 - 0.540i)T \)
61 \( 1 + (0.235 - 0.971i)T \)
71 \( 1 + (0.928 - 0.371i)T \)
73 \( 1 + (0.235 - 0.971i)T \)
79 \( 1 + (-0.0475 + 0.998i)T \)
83 \( 1 + (0.723 - 0.690i)T \)
89 \( 1 + (-0.415 - 0.909i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−22.025594254821696465001991542636, −21.46739336122783792906634188634, −20.76750334024276222179875705842, −19.68145957657981915175971128283, −19.01209772954830282651857757016, −17.94419923984029012837051105253, −17.49814362393448769742038436366, −16.64268414245774948148697029260, −15.629407330083461646977170254497, −14.89227509521744279181159381431, −13.98498983548579432787013440379, −13.22357643082374692948765342451, −12.425240488009908674281362853083, −11.49432608485472481025107963731, −10.63959226159041695584081177747, −9.614751330599916364799944102, −8.867395602200193395986227685, −8.24447505385807186624897329355, −6.86271929746968808859992007699, −6.00335609040226205276636108297, −5.26899178731496441152517868032, −4.40183063322748736897006335360, −2.75440351585360601480414756449, −2.30568702535521691879412139264, −0.84904594928365178013848348207, 1.33473617464412182460090785101, 2.14124772772603153068193874210, 3.33724487291350000853378489332, 4.59244912424267390128221534876, 5.161290272135956934071093182874, 6.628857478690347174893451222940, 6.99166330119299554374904302730, 8.05554735706369297976548442723, 9.36282958003753800419866560842, 9.870736055153069055087776034067, 10.68041857978204828322442244495, 11.63151993401292544209716857401, 12.62383697067767038659116910155, 13.50096801298158708154063885259, 14.25519450809107232413191990564, 14.76746861464454317684625783505, 15.95228023969044030254531554054, 16.947122906722164170105417153164, 17.5445965093340691602614452114, 18.03309750108559917742163835669, 19.242205084279015447517654823363, 19.9902135865445413615889392561, 20.809863581220037867956592669232, 21.43492915179943713209678916988, 22.47337024316514560603800899078

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