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Properties

Label 1-621-621.176-r0-0-0
Degree $1$
Conductor $621$
Sign $-0.816 - 0.577i$
Analytic cond. $2.88391$
Root an. cond. $2.88391$
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  + (−0.997 + 0.0634i)2-s + (0.991 − 0.126i)4-s + (−0.999 + 0.0317i)5-s + (0.266 − 0.963i)7-s + (−0.981 + 0.189i)8-s + (0.995 − 0.0950i)10-s + (0.110 + 0.993i)11-s + (−0.701 + 0.712i)13-s + (−0.204 + 0.978i)14-s + (0.967 − 0.251i)16-s + (−0.888 − 0.458i)17-s + (0.888 − 0.458i)19-s + (−0.987 + 0.158i)20-s + (−0.173 − 0.984i)22-s + (0.997 − 0.0634i)25-s + (0.654 − 0.755i)26-s + ⋯
L(s)  = 1  + (−0.997 + 0.0634i)2-s + (0.991 − 0.126i)4-s + (−0.999 + 0.0317i)5-s + (0.266 − 0.963i)7-s + (−0.981 + 0.189i)8-s + (0.995 − 0.0950i)10-s + (0.110 + 0.993i)11-s + (−0.701 + 0.712i)13-s + (−0.204 + 0.978i)14-s + (0.967 − 0.251i)16-s + (−0.888 − 0.458i)17-s + (0.888 − 0.458i)19-s + (−0.987 + 0.158i)20-s + (−0.173 − 0.984i)22-s + (0.997 − 0.0634i)25-s + (0.654 − 0.755i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (176, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.07077539500 - 0.2226044645i\)
\(L(\frac12)\) \(\approx\) \(0.07077539500 - 0.2226044645i\)
\(L(1)\) \(\approx\) \(0.4898002623 - 0.05499225611i\)
\(L(1)\) \(\approx\) \(0.4898002623 - 0.05499225611i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.997 + 0.0634i)T \)
5 \( 1 + (-0.999 + 0.0317i)T \)
7 \( 1 + (0.266 - 0.963i)T \)
11 \( 1 + (0.110 + 0.993i)T \)
13 \( 1 + (-0.701 + 0.712i)T \)
17 \( 1 + (-0.888 - 0.458i)T \)
19 \( 1 + (0.888 - 0.458i)T \)
29 \( 1 + (0.605 - 0.795i)T \)
31 \( 1 + (-0.987 - 0.158i)T \)
37 \( 1 + (0.786 - 0.618i)T \)
41 \( 1 + (-0.472 + 0.881i)T \)
43 \( 1 + (-0.630 - 0.776i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (0.415 + 0.909i)T \)
59 \( 1 + (-0.967 - 0.251i)T \)
61 \( 1 + (0.0158 + 0.999i)T \)
67 \( 1 + (-0.805 - 0.592i)T \)
71 \( 1 + (-0.235 - 0.971i)T \)
73 \( 1 + (-0.888 + 0.458i)T \)
79 \( 1 + (0.0792 - 0.996i)T \)
83 \( 1 + (0.472 + 0.881i)T \)
89 \( 1 + (-0.327 - 0.945i)T \)
97 \( 1 + (-0.950 - 0.312i)T \)
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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

−23.608625508178247544135841873062, −22.232002238785847111189641456054, −21.73460885370884127538115536599, −20.55654067538819944840856313437, −19.83786311726226601616134782737, −19.199668884218203202386404183304, −18.386972548759056829657740072570, −17.75643085728505964928686538877, −16.5890062741584803981062345513, −15.974028813382441223888626348172, −15.19868347318209753546800919595, −14.51389403805737743141825641611, −12.89420405305372648266695314419, −12.073033687078340474975180314, −11.39578073023870779693774320175, −10.6745914766467971118174708366, −9.509770392227808987412555337944, −8.5566445818368550632186929424, −8.11262889553372067335368056325, −7.12401528291050857280010917959, −6.0442742384551977296703930092, −5.0188252432849522116299489921, −3.46790276113100828535926756393, −2.71926564271505067369099534470, −1.33980323764449664822937740301, 0.17134736394543391423992299305, 1.56186870903654001660415657342, 2.76844969575934060198574186117, 4.08659720227925395676045489426, 4.90743454259990797926981229612, 6.64711730981072155495443909234, 7.2761934494248213743098238312, 7.7898165246387031334204082568, 8.98803588124564852866530463569, 9.76473964542625936765007165241, 10.71941787674456475092377837490, 11.564641303011994773859957313428, 12.09941360771036652309094894007, 13.41995720167921845127839349293, 14.62870119105545737553677594615, 15.26001234635639760173720522650, 16.20934910882325892166825606680, 16.849935563464428530888253238616, 17.758779369494130204792292923441, 18.42437177472156766971803088069, 19.61497727298356008137726857293, 19.95027064229164083635569191592, 20.55505116910715506662759979699, 21.76044483560660564650141245402, 22.87300554237843728273252422688

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