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Properties

Label 1-2205-2205.104-r0-0-0
Degree $1$
Conductor $2205$
Sign $-0.600 + 0.799i$
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  + (0.955 + 0.294i)2-s + (0.826 + 0.563i)4-s + (0.623 + 0.781i)8-s + (−0.955 − 0.294i)11-s + (−0.733 + 0.680i)13-s + (0.365 + 0.930i)16-s + (0.900 + 0.433i)17-s − 19-s + (−0.826 − 0.563i)22-s + (0.826 + 0.563i)23-s + (−0.900 + 0.433i)26-s + (−0.826 + 0.563i)29-s + (0.5 + 0.866i)31-s + (0.0747 + 0.997i)32-s + (0.733 + 0.680i)34-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (0.826 + 0.563i)4-s + (0.623 + 0.781i)8-s + (−0.955 − 0.294i)11-s + (−0.733 + 0.680i)13-s + (0.365 + 0.930i)16-s + (0.900 + 0.433i)17-s − 19-s + (−0.826 − 0.563i)22-s + (0.826 + 0.563i)23-s + (−0.900 + 0.433i)26-s + (−0.826 + 0.563i)29-s + (0.5 + 0.866i)31-s + (0.0747 + 0.997i)32-s + (0.733 + 0.680i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.600 + 0.799i$
Analytic conductor: \(10.2399\)
Root analytic conductor: \(10.2399\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (104, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (0:\ ),\ -0.600 + 0.799i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.000125732 + 2.003260824i\)
\(L(\frac12)\) \(\approx\) \(1.000125732 + 2.003260824i\)
\(L(1)\) \(\approx\) \(1.499306089 + 0.6604623489i\)
\(L(1)\) \(\approx\) \(1.499306089 + 0.6604623489i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
11 \( 1 + (-0.955 - 0.294i)T \)
13 \( 1 + (-0.733 + 0.680i)T \)
17 \( 1 + (0.900 + 0.433i)T \)
19 \( 1 - T \)
23 \( 1 + (0.826 + 0.563i)T \)
29 \( 1 + (-0.826 + 0.563i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.900 + 0.433i)T \)
41 \( 1 + (0.365 - 0.930i)T \)
43 \( 1 + (-0.365 - 0.930i)T \)
47 \( 1 + (-0.955 - 0.294i)T \)
53 \( 1 + (-0.900 + 0.433i)T \)
59 \( 1 + (-0.988 - 0.149i)T \)
61 \( 1 + (-0.826 + 0.563i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (0.900 - 0.433i)T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.733 + 0.680i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
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

−19.61794521938791554270781541785, −18.8804225982973076332902772337, −18.240765871351316739709261813868, −17.1289296902660265912269719843, −16.52630036319323567712174905484, −15.647572004706415925163440851988, −14.88964405140085958659542704661, −14.61241285678807449174754937611, −13.47849333011016945171692285513, −12.854193678321465700613028340948, −12.47091076663283508759562397507, −11.431544277635256947923838775975, −10.83642788725831526336213850074, −10.00774874256825465886183881693, −9.46648052582204897614141400023, −7.92042293604760247962065216914, −7.64581673712313760866106808516, −6.4955209581916763793404056753, −5.820315143323883282317633316989, −4.895347438388875926415163591749, −4.486475699971549591728851924144, −3.20865530897567520462952935031, −2.69433422966239444457458967490, −1.79635391805185859370231759666, −0.478342303780912245704992734317, 1.47103526641229730975855267962, 2.4315263471330722819496646553, 3.199015401242855117270046649909, 4.063438307694900767159134346611, 4.9828952643723841799950850023, 5.50023934560266821045895558181, 6.44773526337845241589140165603, 7.18976694918974317571646840509, 7.90855005984705390474477655937, 8.67958618341033251851374958574, 9.772093522215592606549892562744, 10.69362171748270745427690411321, 11.23441244993190285343772601034, 12.27177709776815156517382874970, 12.6729140835695671145012926325, 13.50569137619018497309905809521, 14.18113514618216548758221739865, 14.95105881913152698738829175761, 15.430208819532349890826469317909, 16.41337749034939766610212311778, 16.85830835733721178958825801878, 17.58783112922664154387634563270, 18.74153914154776492771556518040, 19.260962867003723601005499829778, 20.146262761866921347642209752

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