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Properties

Label 1-2205-2205.1114-r0-0-0
Degree $1$
Conductor $2205$
Sign $0.311 - 0.950i$
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  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.222 − 0.974i)8-s + (0.826 + 0.563i)11-s + (−0.0747 + 0.997i)13-s + (−0.733 + 0.680i)16-s + (−0.623 − 0.781i)17-s + 19-s + (−0.365 − 0.930i)22-s + (−0.365 − 0.930i)23-s + (0.623 − 0.781i)26-s + (0.365 − 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.988 − 0.149i)32-s + (0.0747 + 0.997i)34-s + ⋯
L(s)  = 1  + (−0.826 − 0.563i)2-s + (0.365 + 0.930i)4-s + (0.222 − 0.974i)8-s + (0.826 + 0.563i)11-s + (−0.0747 + 0.997i)13-s + (−0.733 + 0.680i)16-s + (−0.623 − 0.781i)17-s + 19-s + (−0.365 − 0.930i)22-s + (−0.365 − 0.930i)23-s + (0.623 − 0.781i)26-s + (0.365 − 0.930i)29-s + (−0.5 + 0.866i)31-s + (0.988 − 0.149i)32-s + (0.0747 + 0.997i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.311 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $0.311 - 0.950i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (1114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ 0.311 - 0.950i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7993823465 - 0.5790469437i\)
\(L(\frac12)\) \(\approx\) \(0.7993823465 - 0.5790469437i\)
\(L(1)\) \(\approx\) \(0.7339290758 - 0.1881482406i\)
\(L(1)\) \(\approx\) \(0.7339290758 - 0.1881482406i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (-0.826 - 0.563i)T \)
11 \( 1 + (0.826 + 0.563i)T \)
13 \( 1 + (-0.0747 + 0.997i)T \)
17 \( 1 + (-0.623 - 0.781i)T \)
19 \( 1 + T \)
23 \( 1 + (-0.365 - 0.930i)T \)
29 \( 1 + (0.365 - 0.930i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.623 - 0.781i)T \)
41 \( 1 + (-0.733 - 0.680i)T \)
43 \( 1 + (0.733 - 0.680i)T \)
47 \( 1 + (-0.826 - 0.563i)T \)
53 \( 1 + (-0.623 + 0.781i)T \)
59 \( 1 + (0.955 + 0.294i)T \)
61 \( 1 + (0.365 - 0.930i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + (0.623 - 0.781i)T \)
73 \( 1 + (0.900 + 0.433i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (-0.0747 - 0.997i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.77697397892248181494446586297, −19.17330110856838350445608284783, −18.251177331165526509414371404491, −17.69305309338187575583719806168, −17.099555349092921470763893733211, −16.29262828343056668757863761297, −15.67649624644414971288843012694, −14.95276258434244210949572330729, −14.27522175159390115565095229191, −13.49756512127198424231632109724, −12.607650671107353332267529551956, −11.45473118849080272227126648007, −11.14027676387382870718023094492, −10.0331576276004243401644401708, −9.60060376981746530990078050635, −8.59752111494178194003949658043, −8.13210339545858547474257442385, −7.23329725847739378905742284648, −6.47691164350573846424154002941, −5.72045089533320647579665397092, −5.042723283140956729888909653377, −3.80997261647960999223235427572, −2.909874883400862793739328750387, −1.68676252160141170879518632764, −0.941056170885303528641506983882, 0.53240469889272268512262974065, 1.73036482399186154475970573568, 2.305845667973468306518936317086, 3.43201800544621694911937299636, 4.185261393391478086303513932465, 5.03555126603274329805833352424, 6.46014079856865210307746663213, 6.94914584270293338207628822062, 7.70253198056641957728229816462, 8.77880876037874687178945654833, 9.21046775159904753640800074223, 9.926516562967050680094127236205, 10.72951327857345561858906995358, 11.65655293621019640404615298016, 11.9751022207941598129067108787, 12.78668451852141399989193779407, 13.85734067326835268727174177749, 14.3219239558244599105319378569, 15.59118253027069249421606227840, 16.06694946523748491662462828961, 16.9053652627467761664446100911, 17.48997044738064441624408184802, 18.243489000503769794607564784004, 18.812289992505296076243223523151, 19.67650529211484329982323049561

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