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Properties

Label 1-804-804.371-r0-0-0
Degree $1$
Conductor $804$
Sign $-0.811 - 0.584i$
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  + (−0.415 − 0.909i)5-s + (−0.723 + 0.690i)7-s + (−0.995 − 0.0950i)11-s + (0.981 + 0.189i)13-s + (0.888 + 0.458i)17-s + (−0.723 − 0.690i)19-s + (0.928 − 0.371i)23-s + (−0.654 + 0.755i)25-s + (0.5 − 0.866i)29-s + (−0.981 + 0.189i)31-s + (0.928 + 0.371i)35-s + (−0.5 − 0.866i)37-s + (−0.0475 − 0.998i)41-s + (−0.841 + 0.540i)43-s + (−0.786 − 0.618i)47-s + ⋯
L(s)  = 1  + (−0.415 − 0.909i)5-s + (−0.723 + 0.690i)7-s + (−0.995 − 0.0950i)11-s + (0.981 + 0.189i)13-s + (0.888 + 0.458i)17-s + (−0.723 − 0.690i)19-s + (0.928 − 0.371i)23-s + (−0.654 + 0.755i)25-s + (0.5 − 0.866i)29-s + (−0.981 + 0.189i)31-s + (0.928 + 0.371i)35-s + (−0.5 − 0.866i)37-s + (−0.0475 − 0.998i)41-s + (−0.841 + 0.540i)43-s + (−0.786 − 0.618i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $-0.811 - 0.584i$
Analytic conductor: \(3.73376\)
Root analytic conductor: \(3.73376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 804,\ (0:\ ),\ -0.811 - 0.584i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1536422978 - 0.4759474674i\)
\(L(\frac12)\) \(\approx\) \(0.1536422978 - 0.4759474674i\)
\(L(1)\) \(\approx\) \(0.7376040413 - 0.1546627594i\)
\(L(1)\) \(\approx\) \(0.7376040413 - 0.1546627594i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
67 \( 1 \)
good5 \( 1 + (-0.415 - 0.909i)T \)
7 \( 1 + (-0.723 + 0.690i)T \)
11 \( 1 + (-0.995 - 0.0950i)T \)
13 \( 1 + (0.981 + 0.189i)T \)
17 \( 1 + (0.888 + 0.458i)T \)
19 \( 1 + (-0.723 - 0.690i)T \)
23 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (-0.981 + 0.189i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.0475 - 0.998i)T \)
43 \( 1 + (-0.841 + 0.540i)T \)
47 \( 1 + (-0.786 - 0.618i)T \)
53 \( 1 + (-0.841 - 0.540i)T \)
59 \( 1 + (-0.654 - 0.755i)T \)
61 \( 1 + (-0.995 + 0.0950i)T \)
71 \( 1 + (-0.888 + 0.458i)T \)
73 \( 1 + (-0.995 + 0.0950i)T \)
79 \( 1 + (0.327 + 0.945i)T \)
83 \( 1 + (0.580 - 0.814i)T \)
89 \( 1 + (0.142 + 0.989i)T \)
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

−22.82949250303740041216045450384, −21.81471023470612236205387656750, −20.910272496681465208315140981507, −20.21517234109060633508811970593, −19.20441479517552072583936349293, −18.63785300106315114167943729359, −17.98282701486222028247391771742, −16.77130741019781320236592779451, −16.126751736882544317815779140890, −15.31804432735142971127083619403, −14.50963783108133471301455118709, −13.56900811936867930165908462453, −12.91368715687101570353745559722, −11.9011577387005893250821750870, −10.71262793013604903743262622021, −10.525496687682820271005871711302, −9.46397926658104543427844169303, −8.20917404254178357232722909865, −7.482997015580886682179705262665, −6.66622452704042656941973790437, −5.80680885080817047064243686755, −4.58855625485450299169466961344, −3.325877516481644810384604531175, −3.07955933143099084018702077166, −1.42097778871187909119768493478, 0.23196326855319903045533859338, 1.67958880047454038808562230112, 2.93210113162282544164845501907, 3.84202257524798778133340476031, 4.99391935158820213317378852488, 5.71846024322481239388043939399, 6.69996125260108293070538483784, 7.91271702820161352014467075151, 8.63200967320017618963386792625, 9.28925813944987256325803005359, 10.39903206861628819619701295537, 11.26950430026184146449363717246, 12.3059259185316993185133509877, 12.8983584206463022381519871449, 13.488131163101710986321297947896, 14.84522816890682461801157142439, 15.63930549741219128953718768942, 16.173385815815733410193073830396, 16.93540739205973851802630952188, 17.98618717175214116981212161114, 18.94879729257717739134699000360, 19.33679793744557137009586048652, 20.450679525202156414683834042106, 21.15212853373883052791606895142, 21.69253635157198426428516748771

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