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Properties

Label 1-405-405.94-r0-0-0
Degree $1$
Conductor $405$
Sign $0.0193 - 0.999i$
Analytic cond. $1.88081$
Root an. cond. $1.88081$
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  + (−0.597 − 0.802i)2-s + (−0.286 + 0.957i)4-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (0.893 − 0.448i)11-s + (−0.396 − 0.918i)13-s + (−0.993 − 0.116i)14-s + (−0.835 − 0.549i)16-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.893 − 0.448i)22-s + (0.686 + 0.727i)23-s + (−0.5 + 0.866i)26-s + (0.5 + 0.866i)28-s + (−0.993 + 0.116i)29-s + ⋯
L(s)  = 1  + (−0.597 − 0.802i)2-s + (−0.286 + 0.957i)4-s + (0.686 − 0.727i)7-s + (0.939 − 0.342i)8-s + (0.893 − 0.448i)11-s + (−0.396 − 0.918i)13-s + (−0.993 − 0.116i)14-s + (−0.835 − 0.549i)16-s + (−0.766 + 0.642i)17-s + (0.766 + 0.642i)19-s + (−0.893 − 0.448i)22-s + (0.686 + 0.727i)23-s + (−0.5 + 0.866i)26-s + (0.5 + 0.866i)28-s + (−0.993 + 0.116i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0193 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(405\)    =    \(3^{4} \cdot 5\)
Sign: $0.0193 - 0.999i$
Analytic conductor: \(1.88081\)
Root analytic conductor: \(1.88081\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{405} (94, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 405,\ (0:\ ),\ 0.0193 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7326097322 - 0.7185385603i\)
\(L(\frac12)\) \(\approx\) \(0.7326097322 - 0.7185385603i\)
\(L(1)\) \(\approx\) \(0.7832404486 - 0.3955924664i\)
\(L(1)\) \(\approx\) \(0.7832404486 - 0.3955924664i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
good2 \( 1 + (-0.597 - 0.802i)T \)
7 \( 1 + (0.686 - 0.727i)T \)
11 \( 1 + (0.893 - 0.448i)T \)
13 \( 1 + (-0.396 - 0.918i)T \)
17 \( 1 + (-0.766 + 0.642i)T \)
19 \( 1 + (0.766 + 0.642i)T \)
23 \( 1 + (0.686 + 0.727i)T \)
29 \( 1 + (-0.993 + 0.116i)T \)
31 \( 1 + (0.973 - 0.230i)T \)
37 \( 1 + (-0.173 - 0.984i)T \)
41 \( 1 + (0.597 - 0.802i)T \)
43 \( 1 + (0.0581 - 0.998i)T \)
47 \( 1 + (-0.973 - 0.230i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + (0.893 + 0.448i)T \)
61 \( 1 + (-0.286 - 0.957i)T \)
67 \( 1 + (0.993 + 0.116i)T \)
71 \( 1 + (-0.939 - 0.342i)T \)
73 \( 1 + (0.939 - 0.342i)T \)
79 \( 1 + (0.597 + 0.802i)T \)
83 \( 1 + (-0.597 - 0.802i)T \)
89 \( 1 + (-0.939 + 0.342i)T \)
97 \( 1 + (0.835 + 0.549i)T \)
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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

−24.46489613565791193265827776395, −24.21702977737789272159413175688, −22.80845758238141869164095219116, −22.23135286017399635722175576042, −21.04969482868884066145518791589, −19.99210772784139289472679331261, −19.16766003858851854821922276135, −18.263231817339058268240577068574, −17.584365430021797962077737608623, −16.73204723844826179094006534303, −15.80171726894109678772759602544, −14.86725571724377419613602380427, −14.33673608908395319267374594016, −13.23505528955817261006508373396, −11.7556775925273268724455977478, −11.222687675088675980354557319992, −9.70357499538871882416950506584, −9.15003489249556000671443116676, −8.25317626205202880205979255490, −7.08446856665090793180676242058, −6.43336987097490361319101101889, −5.07731410381805589139151677966, −4.46238731448588248629859497629, −2.46964802562703819378790340902, −1.30493299066764559002381381954, 0.8704355036513630798064736305, 1.946669262633439813396899382403, 3.38644234179548126756018560897, 4.16288554942140266673951225405, 5.46724573139952184411490825270, 7.03623198777762649480787611041, 7.856351428776815099816542832464, 8.78062362345588800449801204910, 9.776292370519536809102661257580, 10.74181934856739677431764093441, 11.38154025119807866599401885360, 12.36504571996063354313573014997, 13.37149920356183106454536948627, 14.189815152687283142693070013555, 15.34859569384937854064905986217, 16.63970173934227413156418721301, 17.3168414765919701066006638254, 17.91532472613521213641312382704, 19.085522960362059847153758149590, 19.81507978010379784129823439017, 20.51255020731500667304020397817, 21.35343914587888553301829601230, 22.29269856707444654793941082291, 22.97707264561655923914366937093, 24.32957082088486955535047176905

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