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Properties

Label 1-2205-2205.2-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.644 + 0.764i$
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  + (−0.997 − 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.644 + 0.764i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.644 + 0.764i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8794483321 + 0.4089797324i\)
\(L(\frac12)\) \(\approx\) \(0.8794483321 + 0.4089797324i\)
\(L(1)\) \(\approx\) \(0.7329765985 + 0.05731089500i\)
\(L(1)\) \(\approx\) \(0.7329765985 + 0.05731089500i\)

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 \( 1 + (0.866 - 0.5i)T \)
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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

−19.74474747698765962853052714249, −18.873344499708635697909311257281, −18.12150939762823608261026843593, −17.49669661413460031955262979580, −16.89404510041967858739388276995, −16.14682152514588593248051142287, −15.51246111705945139042043753359, −14.49096644964701125889752061009, −14.25689282801090860108219128279, −12.793078487645599699840241307532, −12.217094316663929100391392292129, −11.42949179092065268732888123641, −10.80228621616505172525227367618, −9.83203306240750835979647215132, −9.26549949036559907662884922899, −8.760376900425520983271690100366, −7.549972362518970563712736675, −7.113676005850958249284649357319, −6.46228297669725006561506919420, −5.29421386782483071647817570776, −4.593565991423920262938876708937, −3.23272672945976044589760893715, −2.54863197620785994306178399114, −1.541481082923278984305314468753, −0.54800473167562991624142238552, 0.96445716217359411373124331837, 1.740696440093166458718432753728, 2.79211915540413535768745073247, 3.585549703312550776279185764612, 4.60975368532359469559524171168, 5.96227555249450866832612476064, 6.2480286489777705280808750797, 7.5167702932842287842548958390, 7.81554568691330736464604434707, 8.82930098136917455424160177853, 9.56582299789524092881702222225, 10.03078588999018509784386571123, 11.06704266100497409525174729669, 11.59288429175736446319934202838, 12.36266741505998573980889385623, 13.09934755798902557250521192569, 14.31195860188269474505001522245, 14.84663210790416341847437781638, 15.575906673466905622352761051, 16.64251916988655014997901232796, 16.880444349558301929418265890944, 17.62528412937300846123335031271, 18.41122037429425186754615708124, 19.3199163274188345977458442256, 19.4940302094060745881843116357

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