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Properties

Label 1-2205-2205.1418-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.660 + 0.750i$
Analytic cond. $10.2399$
Root an. cond. $10.2399$
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  + (0.294 + 0.955i)2-s + (−0.826 + 0.563i)4-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)11-s + (0.294 + 0.955i)13-s + (0.365 − 0.930i)16-s + (−0.563 + 0.826i)17-s + (0.5 − 0.866i)19-s + (0.997 − 0.0747i)22-s + (−0.433 + 0.900i)23-s + (−0.826 + 0.563i)26-s + (0.0747 − 0.997i)29-s + (−0.5 + 0.866i)31-s + (0.997 + 0.0747i)32-s + (−0.955 − 0.294i)34-s + ⋯
L(s)  = 1  + (0.294 + 0.955i)2-s + (−0.826 + 0.563i)4-s + (−0.781 − 0.623i)8-s + (0.222 − 0.974i)11-s + (0.294 + 0.955i)13-s + (0.365 − 0.930i)16-s + (−0.563 + 0.826i)17-s + (0.5 − 0.866i)19-s + (0.997 − 0.0747i)22-s + (−0.433 + 0.900i)23-s + (−0.826 + 0.563i)26-s + (0.0747 − 0.997i)29-s + (−0.5 + 0.866i)31-s + (0.997 + 0.0747i)32-s + (−0.955 − 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.660 + 0.750i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.660 + 0.750i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1418, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.660 + 0.750i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5939050662 + 1.313634249i\)
\(L(\frac12)\) \(\approx\) \(0.5939050662 + 1.313634249i\)
\(L(1)\) \(\approx\) \(0.8992359408 + 0.6056761146i\)
\(L(1)\) \(\approx\) \(0.8992359408 + 0.6056761146i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.294 + 0.955i)T \)
11 \( 1 + (0.222 - 0.974i)T \)
13 \( 1 + (0.294 + 0.955i)T \)
17 \( 1 + (-0.563 + 0.826i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.433 + 0.900i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.997 - 0.0747i)T \)
41 \( 1 + (0.988 + 0.149i)T \)
43 \( 1 + (-0.149 - 0.988i)T \)
47 \( 1 + (0.294 + 0.955i)T \)
53 \( 1 + (0.997 - 0.0747i)T \)
59 \( 1 + (-0.988 + 0.149i)T \)
61 \( 1 + (0.826 + 0.563i)T \)
67 \( 1 + (0.866 + 0.5i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.680 - 0.733i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.294 - 0.955i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
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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

−19.75810216540263628459874061063, −18.68638214224739314666504272176, −18.15757246839283789258319690672, −17.66104352927768080838665528064, −16.64477391382992046545127025626, −15.7193595804807405851211409572, −14.94603972020831942938023074742, −14.29374465115357583169605686633, −13.569497602383266612176245748554, −12.67576180955503915387662891464, −12.31622852390473636139267569847, −11.4337984263524366336017004515, −10.67068206036341729734611715988, −10.008863520550730993114594841328, −9.34447926818203026212213230063, −8.50550391947755900555615301209, −7.62898630862211062633772661287, −6.63625910264520913842043398540, −5.64865242629643256812403328186, −4.96485011350922246353684914743, −4.126875003117406659050409115682, −3.34060628577522131996120798228, −2.43185221840075633039550797274, −1.64895904769047514310044158935, −0.52283926856941538624896969566, 0.97862028129709451664263462211, 2.29505506618506277724766184371, 3.51892331537612248546228991532, 4.007930056228088953100355652794, 4.9996920818677921912660070970, 5.81983058374829694854556565595, 6.464814416180141825525387532290, 7.20185659402305625828418978913, 8.05122828892665827549104768648, 8.91064782870388341289115158416, 9.23601849500945637742375043355, 10.42251912787368754798577081952, 11.37086028153039485252460583898, 11.996157537212155788743869055576, 12.99913012323064538745544421052, 13.69933046057394035695269821172, 14.08195748553803116306448753086, 14.99447325490149517528273437005, 15.85854092366903536477653099064, 16.12361403554569598375260360093, 17.19196470676085132843051325641, 17.523611349984192904949502843784, 18.47860708105247413882589678939, 19.17196758851669030810886963133, 19.81156633952114377653566414716

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