↑

Properties

Label 1-2205-2205.1102-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.972 + 0.234i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.997 + 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.974 + 0.222i)8-s + (−0.900 − 0.433i)11-s + (−0.997 + 0.0747i)13-s + (0.955 − 0.294i)16-s + (−0.149 + 0.988i)17-s + (−0.5 + 0.866i)19-s + (0.930 + 0.365i)22-s + (0.781 − 0.623i)23-s + (0.988 − 0.149i)26-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (−0.930 + 0.365i)32-s + (0.0747 − 0.997i)34-s + ⋯
L(s)  = 1  + (−0.997 + 0.0747i)2-s + (0.988 − 0.149i)4-s + (−0.974 + 0.222i)8-s + (−0.900 − 0.433i)11-s + (−0.997 + 0.0747i)13-s + (0.955 − 0.294i)16-s + (−0.149 + 0.988i)17-s + (−0.5 + 0.866i)19-s + (0.930 + 0.365i)22-s + (0.781 − 0.623i)23-s + (0.988 − 0.149i)26-s + (−0.365 − 0.930i)29-s + (0.5 − 0.866i)31-s + (−0.930 + 0.365i)32-s + (0.0747 − 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.972 + 0.234i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.972 + 0.234i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1102, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.972 + 0.234i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6934127583 + 0.08240345057i\)
\(L(\frac12)\) \(\approx\) \(0.6934127583 + 0.08240345057i\)
\(L(1)\) \(\approx\) \(0.6073599894 + 0.02140241288i\)
\(L(1)\) \(\approx\) \(0.6073599894 + 0.02140241288i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.997 + 0.0747i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (-0.997 + 0.0747i)T \)
17 \( 1 + (-0.149 + 0.988i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (-0.365 - 0.930i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.930 + 0.365i)T \)
41 \( 1 + (0.733 - 0.680i)T \)
43 \( 1 + (-0.680 + 0.733i)T \)
47 \( 1 + (0.997 - 0.0747i)T \)
53 \( 1 + (-0.930 - 0.365i)T \)
59 \( 1 + (-0.733 - 0.680i)T \)
61 \( 1 + (0.988 + 0.149i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (0.623 + 0.781i)T \)
73 \( 1 + (-0.563 + 0.826i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.997 + 0.0747i)T \)
89 \( 1 + (0.0747 - 0.997i)T \)
97 \( 1 + (0.866 + 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−19.56076536181907864911261724437, −19.04628721923752658265521745399, −18.11490954647188559981383230709, −17.68240584198161319561098027005, −16.98299643989716992354247856640, −16.16979732607807834823183958705, −15.47365356457918749038564187566, −14.96463333267073269920767090132, −13.93807295397775562661321165091, −12.949744795971767586332634920750, −12.3087910394489854948687231192, −11.52269804076657432796433125531, −10.69918572311509086605952039284, −10.18032979058342395158679194730, −9.22400231365430572058551561996, −8.83685289301408216328647097244, −7.60066975088072680215638705365, −7.31779093988675177978652226750, −6.50242585336459662574062074824, −5.28941443132112398699451584929, −4.7485225406817363442981425663, −3.2423163887301165583097144276, −2.625459579930102797521900507851, −1.78595349040958620590913442881, −0.53691099283672187703513421680, 0.59985496466975858164793358314, 1.9021197660635948848990615997, 2.51726518326074805134761488781, 3.51446648043569416147331072817, 4.66647879015731319412803094917, 5.68928111768245798358608217080, 6.326011521580432354206183746627, 7.25978024186954817820229218100, 8.015931445031783363199956432823, 8.49810120442491778121594732241, 9.48090033649443600889723664187, 10.19289447546255529312443663868, 10.73519589786442684400003009112, 11.53175462190261458740378162336, 12.427076265802950528591077433776, 12.99387820881669137104838694244, 14.15197373554996843299274164915, 14.95440495583683257164729984231, 15.46835348598718807290759697367, 16.31948469333991326798297860671, 17.1421298550474249735421336816, 17.34197544519387613988206872170, 18.51485399637829525317751707140, 18.98252067656826718630789324902, 19.45467227865801032781967470052

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