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Properties

Label 1-621-621.218-r0-0-0
Degree $1$
Conductor $621$
Sign $-0.760 + 0.649i$
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  + (0.701 − 0.712i)2-s + (−0.0158 − 0.999i)4-s + (−0.386 − 0.922i)5-s + (−0.527 − 0.849i)7-s + (−0.723 − 0.690i)8-s + (−0.928 − 0.371i)10-s + (−0.823 + 0.567i)11-s + (0.472 + 0.881i)13-s + (−0.975 − 0.220i)14-s + (−0.999 + 0.0317i)16-s + (−0.327 − 0.945i)17-s + (0.327 − 0.945i)19-s + (−0.916 + 0.400i)20-s + (−0.173 + 0.984i)22-s + (−0.701 + 0.712i)25-s + (0.959 + 0.281i)26-s + ⋯
L(s)  = 1  + (0.701 − 0.712i)2-s + (−0.0158 − 0.999i)4-s + (−0.386 − 0.922i)5-s + (−0.527 − 0.849i)7-s + (−0.723 − 0.690i)8-s + (−0.928 − 0.371i)10-s + (−0.823 + 0.567i)11-s + (0.472 + 0.881i)13-s + (−0.975 − 0.220i)14-s + (−0.999 + 0.0317i)16-s + (−0.327 − 0.945i)17-s + (0.327 − 0.945i)19-s + (−0.916 + 0.400i)20-s + (−0.173 + 0.984i)22-s + (−0.701 + 0.712i)25-s + (0.959 + 0.281i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 621 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(621\)    =    \(3^{3} \cdot 23\)
Sign: $-0.760 + 0.649i$
Analytic conductor: \(2.88391\)
Root analytic conductor: \(2.88391\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{621} (218, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 621,\ (0:\ ),\ -0.760 + 0.649i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3305026762 - 0.8961785149i\)
\(L(\frac12)\) \(\approx\) \(-0.3305026762 - 0.8961785149i\)
\(L(1)\) \(\approx\) \(0.7145047037 - 0.7860152985i\)
\(L(1)\) \(\approx\) \(0.7145047037 - 0.7860152985i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.701 - 0.712i)T \)
5 \( 1 + (-0.386 - 0.922i)T \)
7 \( 1 + (-0.527 - 0.849i)T \)
11 \( 1 + (-0.823 + 0.567i)T \)
13 \( 1 + (0.472 + 0.881i)T \)
17 \( 1 + (-0.327 - 0.945i)T \)
19 \( 1 + (0.327 - 0.945i)T \)
29 \( 1 + (-0.873 + 0.486i)T \)
31 \( 1 + (-0.916 - 0.400i)T \)
37 \( 1 + (0.888 - 0.458i)T \)
41 \( 1 + (0.605 - 0.795i)T \)
43 \( 1 + (-0.110 + 0.993i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.142 + 0.989i)T \)
59 \( 1 + (0.999 + 0.0317i)T \)
61 \( 1 + (0.553 - 0.832i)T \)
67 \( 1 + (0.0792 - 0.996i)T \)
71 \( 1 + (-0.580 - 0.814i)T \)
73 \( 1 + (-0.327 + 0.945i)T \)
79 \( 1 + (0.204 - 0.978i)T \)
83 \( 1 + (-0.605 - 0.795i)T \)
89 \( 1 + (0.235 + 0.971i)T \)
97 \( 1 + (-0.678 - 0.734i)T \)
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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

−23.42372143438752554372342438319, −22.68233431036430644331262891947, −22.07492711401590061143349760180, −21.37063742921741309517007839996, −20.39044314089122239501657095735, −19.21104263657387316952680358508, −18.38951623001691854690700719624, −17.8388267915991155351565727618, −16.48340389728318598343271841423, −15.877634189320891291538516748923, −15.07492008114514020956841886019, −14.615582807633554092563743243441, −13.3203915783331894737579570428, −12.85027894876599563044733542479, −11.776061435028565725388438971607, −10.95040815977860148094334843444, −9.90929336642181491661786942445, −8.4785697290884686966163763301, −7.98144524593511246039175779405, −6.90159637281688235278833453466, −5.90796124681908251001550981544, −5.50752910475790101780715278969, −3.88552781258247549362785871865, −3.23745164897700551737475880534, −2.33319131452907692333625604580, 0.36024820257343875969055375729, 1.60397062256379085979651611100, 2.84391390999479861989721363682, 3.9977006903629058726156048942, 4.61880781825463635928703364893, 5.519879106747258724958970075121, 6.79706760964908776104162159271, 7.63389234930061789968102187775, 9.20542293502891559565544928793, 9.54851839866457192160889971428, 10.8868411914349785514801427534, 11.42007962156937689254790618704, 12.556426552088134231673152203789, 13.15081565759697855678302927037, 13.74718045497409810832767585028, 14.84924601795312125381778200169, 15.98913250319478495696188916577, 16.27157984130469926639025331110, 17.62701788806300891605928433929, 18.58289005199787355301809072390, 19.50967470979343205748097730389, 20.285269650260454511904384896050, 20.6152724284218517129017547223, 21.5791839986112538387357430453, 22.5832947128775578606487144919

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