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Properties

Label 1-2205-2205.137-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.980 - 0.196i$
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  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (−0.955 − 0.294i)11-s + (0.294 − 0.955i)13-s + (0.623 − 0.781i)16-s + (0.997 − 0.0747i)17-s + (0.5 − 0.866i)19-s + (0.997 + 0.0747i)22-s + (−0.563 + 0.826i)23-s + (−0.0747 + 0.997i)26-s + (0.0747 + 0.997i)29-s + 31-s + (−0.433 + 0.900i)32-s + (−0.955 + 0.294i)34-s + ⋯
L(s)  = 1  + (−0.974 + 0.222i)2-s + (0.900 − 0.433i)4-s + (−0.781 + 0.623i)8-s + (−0.955 − 0.294i)11-s + (0.294 − 0.955i)13-s + (0.623 − 0.781i)16-s + (0.997 − 0.0747i)17-s + (0.5 − 0.866i)19-s + (0.997 + 0.0747i)22-s + (−0.563 + 0.826i)23-s + (−0.0747 + 0.997i)26-s + (0.0747 + 0.997i)29-s + 31-s + (−0.433 + 0.900i)32-s + (−0.955 + 0.294i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.196i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.980 - 0.196i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (137, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.980 - 0.196i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9728795982 - 0.09662399881i\)
\(L(\frac12)\) \(\approx\) \(0.9728795982 - 0.09662399881i\)
\(L(1)\) \(\approx\) \(0.7244022356 + 0.004839642885i\)
\(L(1)\) \(\approx\) \(0.7244022356 + 0.004839642885i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (0.294 - 0.955i)T \)
17 \( 1 + (0.997 - 0.0747i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.563 + 0.826i)T \)
29 \( 1 + (0.0747 + 0.997i)T \)
31 \( 1 + T \)
37 \( 1 + (0.563 + 0.826i)T \)
41 \( 1 + (0.988 - 0.149i)T \)
43 \( 1 + (-0.149 + 0.988i)T \)
47 \( 1 + (-0.974 + 0.222i)T \)
53 \( 1 + (-0.563 + 0.826i)T \)
59 \( 1 + (0.623 - 0.781i)T \)
61 \( 1 + (-0.900 - 0.433i)T \)
67 \( 1 + iT \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.294 - 0.955i)T \)
79 \( 1 - T \)
83 \( 1 + (0.294 + 0.955i)T \)
89 \( 1 + (-0.733 - 0.680i)T \)
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

−19.62076742362658915917292966561, −18.85035873385074831247411118563, −18.45286957383574652647968100094, −17.73418791791634775827105754837, −16.868678401568649118115353308047, −16.2482809011009524626280684215, −15.75457202167863125474548983260, −14.77072597050333271840297331647, −14.03114383647550077246683426039, −13.027033543877487343981485923442, −12.220729727788440713354167296833, −11.69510282695334856563650997859, −10.80077380611959713811380854456, −10.0236733526485132477634923854, −9.639691876884015872354492951260, −8.549521516959154793355019594481, −7.959388268783513988456923830, −7.32111437345254346259464422799, −6.33580660284369355658835530719, −5.6719466734505895179626589149, −4.44209116285241561426155848617, −3.54264155462747581148654925370, −2.55733823898255272506003859870, −1.84288684793044682584701833305, −0.757314308877762793336319742483, 0.669239073321203464537638173254, 1.528664160105686536459564441234, 2.868509441006975398014914714518, 3.16232731077955170923161800453, 4.80012578828173558097156996428, 5.5684115120972594485940754150, 6.21459304389558105486951318149, 7.28149801000768660141486471434, 7.9156066006560815470312179378, 8.396705074093931911916195335703, 9.505705055916518049171126231100, 9.98907641825959410244053014972, 10.82079099599381885099195287275, 11.39256900597768127409744297116, 12.316489335992621529554284832, 13.13680084239089689759580400505, 14.00719591166109852212423361988, 14.863812504232164168585424128448, 15.71178853329393419850163852675, 15.99277514527539764257399796916, 16.89472054004222324341117513568, 17.75457409966968575019794097618, 18.13273855627724112168038366777, 18.87182108726789742361009294697, 19.65853048047914568749611857955

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