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Properties

Label 1-621-621.203-r0-0-0
Degree $1$
Conductor $621$
Sign $-0.877 + 0.479i$
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  + (−0.356 + 0.934i)2-s + (−0.745 − 0.666i)4-s + (−0.823 + 0.567i)5-s + (0.916 + 0.400i)7-s + (0.888 − 0.458i)8-s + (−0.235 − 0.971i)10-s + (−0.857 + 0.513i)11-s + (0.805 + 0.592i)13-s + (−0.701 + 0.712i)14-s + (0.110 + 0.993i)16-s + (0.928 − 0.371i)17-s + (−0.928 − 0.371i)19-s + (0.991 + 0.126i)20-s + (−0.173 − 0.984i)22-s + (0.356 − 0.934i)25-s + (−0.841 + 0.540i)26-s + ⋯
L(s)  = 1  + (−0.356 + 0.934i)2-s + (−0.745 − 0.666i)4-s + (−0.823 + 0.567i)5-s + (0.916 + 0.400i)7-s + (0.888 − 0.458i)8-s + (−0.235 − 0.971i)10-s + (−0.857 + 0.513i)11-s + (0.805 + 0.592i)13-s + (−0.701 + 0.712i)14-s + (0.110 + 0.993i)16-s + (0.928 − 0.371i)17-s + (−0.928 − 0.371i)19-s + (0.991 + 0.126i)20-s + (−0.173 − 0.984i)22-s + (0.356 − 0.934i)25-s + (−0.841 + 0.540i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $-0.877 + 0.479i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (203, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ -0.877 + 0.479i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2138072055 + 0.8377945597i\)
\(L(\frac12)\) \(\approx\) \(0.2138072055 + 0.8377945597i\)
\(L(1)\) \(\approx\) \(0.6113419870 + 0.4903183271i\)
\(L(1)\) \(\approx\) \(0.6113419870 + 0.4903183271i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (-0.356 + 0.934i)T \)
5 \( 1 + (-0.823 + 0.567i)T \)
7 \( 1 + (0.916 + 0.400i)T \)
11 \( 1 + (-0.857 + 0.513i)T \)
13 \( 1 + (0.805 + 0.592i)T \)
17 \( 1 + (0.928 - 0.371i)T \)
19 \( 1 + (-0.928 - 0.371i)T \)
29 \( 1 + (0.204 + 0.978i)T \)
31 \( 1 + (0.991 - 0.126i)T \)
37 \( 1 + (0.995 - 0.0950i)T \)
41 \( 1 + (0.0792 + 0.996i)T \)
43 \( 1 + (0.386 + 0.922i)T \)
47 \( 1 + (-0.766 - 0.642i)T \)
53 \( 1 + (-0.959 + 0.281i)T \)
59 \( 1 + (-0.110 + 0.993i)T \)
61 \( 1 + (-0.296 - 0.954i)T \)
67 \( 1 + (-0.873 + 0.486i)T \)
71 \( 1 + (-0.981 - 0.189i)T \)
73 \( 1 + (0.928 + 0.371i)T \)
79 \( 1 + (-0.997 + 0.0634i)T \)
83 \( 1 + (-0.0792 + 0.996i)T \)
89 \( 1 + (0.0475 + 0.998i)T \)
97 \( 1 + (-0.967 + 0.251i)T \)
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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

−22.753292005826747535553735866260, −21.348990033342143942358865314269, −20.90573694769615608319189357819, −20.37888666455044360914372489591, −19.25615190458510111579334428418, −18.79832119404463706657162684003, −17.753399509613426526515258178585, −17.018939818972445947049183550428, −16.15471671232020406629985136135, −15.171953007376250364779097254781, −14.03350841464763533841561207088, −13.16358721776926983108681248862, −12.41289823567984843273825510122, −11.50918484045458593994744412624, −10.8082770716039476515500924869, −10.116147740310310407271557020300, −8.70457289482510596618488244364, −8.11066924456818264933461589001, −7.65822198033980348797630131555, −5.79975550682876504236209162382, −4.71097039813095493475159722529, −3.952309244491645593676685265559, −2.97516810794338634660378304319, −1.58598798687598194339870555026, −0.5714961947072926583547508071, 1.30883895257872884245439180878, 2.77517671098345243502511909982, 4.24681018338899590376307662971, 4.90921196192218195346682037778, 6.07358633631942376949925343691, 6.994679510515042812735372016191, 7.94084975480549003188035528097, 8.35081977939741097212185086292, 9.51957521561144347199606200766, 10.61778877515839804163944023118, 11.28038175330731149608534493493, 12.39369452040077340832409888909, 13.51840198195611568233596327114, 14.55210079438198561139291667728, 14.97772744892271319719014626670, 15.84558283952018534003763716848, 16.51365075225575304153602805136, 17.70488785139620492446801075077, 18.34039992948696553383796802393, 18.862854436931586848930869892203, 19.82763543164883513194165284975, 20.96113398218443672068272833162, 21.75335822398593200965778471662, 23.002955744385980663509410193892, 23.40907088239752438924710701909

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