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Properties

Label 1-2205-2205.2029-r1-0-0
Degree $1$
Conductor $2205$
Sign $-0.575 + 0.818i$
Analytic cond. $236.960$
Root an. cond. $236.960$
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  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (0.900 − 0.433i)8-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)13-s + (0.826 − 0.563i)16-s + (−0.222 − 0.974i)17-s − 19-s + (−0.955 + 0.294i)22-s + (−0.955 + 0.294i)23-s + (0.222 − 0.974i)26-s + (0.955 + 0.294i)29-s + (0.5 + 0.866i)31-s + (0.733 − 0.680i)32-s + (−0.365 − 0.930i)34-s + ⋯
L(s)  = 1  + (0.988 − 0.149i)2-s + (0.955 − 0.294i)4-s + (0.900 − 0.433i)8-s + (−0.988 + 0.149i)11-s + (0.365 − 0.930i)13-s + (0.826 − 0.563i)16-s + (−0.222 − 0.974i)17-s − 19-s + (−0.955 + 0.294i)22-s + (−0.955 + 0.294i)23-s + (0.222 − 0.974i)26-s + (0.955 + 0.294i)29-s + (0.5 + 0.866i)31-s + (0.733 − 0.680i)32-s + (−0.365 − 0.930i)34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2205\)    =    \(3^{2} \cdot 5 \cdot 7^{2}\)
Sign: $-0.575 + 0.818i$
Analytic conductor: \(236.960\)
Root analytic conductor: \(236.960\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2205} (2029, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2205,\ (1:\ ),\ -0.575 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3250880545 + 0.6258483921i\)
\(L(\frac12)\) \(\approx\) \(0.3250880545 + 0.6258483921i\)
\(L(1)\) \(\approx\) \(1.555445260 - 0.2241657882i\)
\(L(1)\) \(\approx\) \(1.555445260 - 0.2241657882i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good2 \( 1 + (0.988 - 0.149i)T \)
11 \( 1 + (-0.988 + 0.149i)T \)
13 \( 1 + (0.365 - 0.930i)T \)
17 \( 1 + (-0.222 - 0.974i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.955 + 0.294i)T \)
29 \( 1 + (0.955 + 0.294i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (0.222 + 0.974i)T \)
41 \( 1 + (-0.826 - 0.563i)T \)
43 \( 1 + (-0.826 + 0.563i)T \)
47 \( 1 + (-0.988 + 0.149i)T \)
53 \( 1 + (0.222 - 0.974i)T \)
59 \( 1 + (-0.0747 - 0.997i)T \)
61 \( 1 + (-0.955 - 0.294i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.222 + 0.974i)T \)
73 \( 1 + (0.623 - 0.781i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + (0.365 + 0.930i)T \)
89 \( 1 + (-0.623 + 0.781i)T \)
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

−19.45920074196403504793076705346, −18.67054997999145801882946059454, −17.81159054135390026643322199912, −16.81989133860939282140203289826, −16.405781764595234818318061433459, −15.4400366858079459527393985149, −15.08193661607572209397820391280, −14.07799789388409263716692335925, −13.5460412114995208848951503925, −12.83671631090581341727367556968, −12.14287551521253555264017986012, −11.35704404029491799062001068713, −10.61236814760136655779505668136, −9.99943717644929888786975625536, −8.609691031677685351233173478059, −8.12481597923259521990124415858, −7.17906236061303282505148532789, −6.229848685277729219616546412908, −5.920211133017054794761016589937, −4.66448184207052868907988926356, −4.25033930494964126627076399778, −3.28164085430850259286281929966, −2.304347628888337452899658905769, −1.6754822574526841775846559284, −0.07257672474234893348049144999, 1.1412809740862552856857038564, 2.24706355441939075700368373505, 2.94481634610326826857106479013, 3.721190881055513809602322075125, 4.88484621604748869750128598266, 5.15307310813229989865749248147, 6.27197143080872735437385089996, 6.82337245873930969684472014754, 7.91160602955996815716066131338, 8.400550351946018903209127116, 9.866364482661040860753679450656, 10.32738793047458922709130655038, 11.10603809876431104997795559928, 11.89165222455404826908202373915, 12.639308927196654725560476918, 13.27027487890545713738901127077, 13.86657044971160401627254423035, 14.68973228940931157540993443444, 15.578445023186234786263685559284, 15.80917834857677264928887082440, 16.71777472423155292618857303749, 17.75125507865257973591598481783, 18.33771441941557642462704495036, 19.28401606198982972813580464518, 20.04495769826733664317176878198

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