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Properties

Label 1-2205-2205.1103-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: $\chi_{2205} (1103, \cdot )$
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

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

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