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Properties

Label 1-804-804.47-r0-0-0
Degree $1$
Conductor $804$
Sign $0.547 - 0.836i$
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  + (0.654 − 0.755i)5-s + (0.888 − 0.458i)7-s + (−0.327 − 0.945i)11-s + (−0.786 + 0.618i)13-s + (0.995 − 0.0950i)17-s + (0.888 + 0.458i)19-s + (0.235 + 0.971i)23-s + (−0.142 − 0.989i)25-s + (0.5 − 0.866i)29-s + (0.786 + 0.618i)31-s + (0.235 − 0.971i)35-s + (−0.5 − 0.866i)37-s + (−0.580 − 0.814i)41-s + (−0.415 + 0.909i)43-s + (0.723 − 0.690i)47-s + ⋯
L(s)  = 1  + (0.654 − 0.755i)5-s + (0.888 − 0.458i)7-s + (−0.327 − 0.945i)11-s + (−0.786 + 0.618i)13-s + (0.995 − 0.0950i)17-s + (0.888 + 0.458i)19-s + (0.235 + 0.971i)23-s + (−0.142 − 0.989i)25-s + (0.5 − 0.866i)29-s + (0.786 + 0.618i)31-s + (0.235 − 0.971i)35-s + (−0.5 − 0.866i)37-s + (−0.580 − 0.814i)41-s + (−0.415 + 0.909i)43-s + (0.723 − 0.690i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.547 - 0.836i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(804\)    =    \(2^{2} \cdot 3 \cdot 67\)
Sign: $0.547 - 0.836i$
Analytic conductor: \(3.73376\)
Root analytic conductor: \(3.73376\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{804} (47, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 804,\ (0:\ ),\ 0.547 - 0.836i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.599055714 - 0.8642305451i\)
\(L(\frac12)\) \(\approx\) \(1.599055714 - 0.8642305451i\)
\(L(1)\) \(\approx\) \(1.274420262 - 0.3148485197i\)
\(L(1)\) \(\approx\) \(1.274420262 - 0.3148485197i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
67 \( 1 \)
good5 \( 1 + (0.654 - 0.755i)T \)
7 \( 1 + (0.888 - 0.458i)T \)
11 \( 1 + (-0.327 - 0.945i)T \)
13 \( 1 + (-0.786 + 0.618i)T \)
17 \( 1 + (0.995 - 0.0950i)T \)
19 \( 1 + (0.888 + 0.458i)T \)
23 \( 1 + (0.235 + 0.971i)T \)
29 \( 1 + (0.5 - 0.866i)T \)
31 \( 1 + (0.786 + 0.618i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + (-0.580 - 0.814i)T \)
43 \( 1 + (-0.415 + 0.909i)T \)
47 \( 1 + (0.723 - 0.690i)T \)
53 \( 1 + (-0.415 - 0.909i)T \)
59 \( 1 + (-0.142 + 0.989i)T \)
61 \( 1 + (-0.327 + 0.945i)T \)
71 \( 1 + (-0.995 - 0.0950i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (-0.928 - 0.371i)T \)
83 \( 1 + (0.981 + 0.189i)T \)
89 \( 1 + (0.959 - 0.281i)T \)
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

−22.2302277111169403382538890153, −21.73684250900019680302604602105, −20.698309329899646722040476614888, −20.251526811053669183978085942, −18.881544490773799965234594916087, −18.38822985493122203633692588362, −17.547939850079194267186320414323, −17.09026712805971816482748197246, −15.69905774473658601421181189840, −14.92906661856005084447700450045, −14.44075505790083445134973774187, −13.54847086410486997007606065260, −12.43915437555377427502335362071, −11.82678288822347690601597754497, −10.66182873879171448593289129548, −10.10240896280308380281895257085, −9.24457116262811225006453872425, −8.02642312233064985459809273434, −7.36486597375756813026795397545, −6.37876695067381410038760162593, −5.25259003633461625793292316087, −4.77587371919818459098542722084, −3.12151659543091379184348272734, −2.44215787698556100879884545261, −1.36890528282402939901673544255, 0.93700196300246786018442032524, 1.804447440869198664846903074049, 3.07684378868285831383150053555, 4.28439093029489026303399557445, 5.23246825339437763449892018403, 5.75945074434432478238768138482, 7.14201850768665135974595879671, 7.97278407379814156254610274397, 8.77435285122686882912582756283, 9.767319200054687671733493824388, 10.43129123480239004358426207058, 11.65879394652889366616613125473, 12.11798858880798068734547849749, 13.423762014450897941556635330957, 13.90045685134136549430498094089, 14.59215180754815019235850433158, 15.86669090499464880253238990108, 16.58775331208463090085835042980, 17.27160792135138473975737011797, 17.9459443759641264025362517948, 18.97428889746139340183755878405, 19.746841068772317293674497457577, 20.81870363772447430725806333257, 21.190404698528967374777579944793, 21.84038846709773746767110684604

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