Properties

Label 1-435-435.158-r0-0-0
Degree $1$
Conductor $435$
Sign $-0.944 - 0.329i$
Analytic cond. $2.02013$
Root an. cond. $2.02013$
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  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.974 + 0.222i)7-s + (−0.433 − 0.900i)8-s + (−0.900 − 0.433i)11-s + (0.433 − 0.900i)13-s + (−0.623 + 0.781i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (−0.222 + 0.974i)19-s + (−0.974 + 0.222i)22-s + (−0.781 − 0.623i)23-s + (−0.222 − 0.974i)26-s + i·28-s + (−0.623 − 0.781i)31-s + (−0.974 + 0.222i)32-s + ⋯
L(s)  = 1  + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.974 + 0.222i)7-s + (−0.433 − 0.900i)8-s + (−0.900 − 0.433i)11-s + (0.433 − 0.900i)13-s + (−0.623 + 0.781i)14-s + (−0.900 − 0.433i)16-s − i·17-s + (−0.222 + 0.974i)19-s + (−0.974 + 0.222i)22-s + (−0.781 − 0.623i)23-s + (−0.222 − 0.974i)26-s + i·28-s + (−0.623 − 0.781i)31-s + (−0.974 + 0.222i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(435\)    =    \(3 \cdot 5 \cdot 29\)
Sign: $-0.944 - 0.329i$
Analytic conductor: \(2.02013\)
Root analytic conductor: \(2.02013\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{435} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 435,\ (0:\ ),\ -0.944 - 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2072563421 - 1.222667188i\)
\(L(\frac12)\) \(\approx\) \(0.2072563421 - 1.222667188i\)
\(L(1)\) \(\approx\) \(0.9683720687 - 0.7212053788i\)
\(L(1)\) \(\approx\) \(0.9683720687 - 0.7212053788i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.781 - 0.623i)T \)
7 \( 1 + (-0.974 + 0.222i)T \)
11 \( 1 + (-0.900 - 0.433i)T \)
13 \( 1 + (0.433 - 0.900i)T \)
17 \( 1 - iT \)
19 \( 1 + (-0.222 + 0.974i)T \)
23 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + (-0.623 - 0.781i)T \)
37 \( 1 + (-0.433 - 0.900i)T \)
41 \( 1 + T \)
43 \( 1 + (-0.781 - 0.623i)T \)
47 \( 1 + (-0.433 + 0.900i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + T \)
61 \( 1 + (0.222 + 0.974i)T \)
67 \( 1 + (0.433 + 0.900i)T \)
71 \( 1 + (0.900 + 0.433i)T \)
73 \( 1 + (-0.781 - 0.623i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (0.974 + 0.222i)T \)
89 \( 1 + (-0.623 - 0.781i)T \)
97 \( 1 + (-0.974 - 0.222i)T \)
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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

−24.244309629009103392934926721314, −23.53522471550925806541798789526, −23.04052645427374351308406338293, −21.85332990056448580594877910925, −21.446921498335455360077994901, −20.28340762141694142592469203716, −19.46912332308504214483205679476, −18.291824785388485971422520620068, −17.35514601240379535324374893736, −16.41915551142361026555361537493, −15.76391990353528026231560122011, −14.99398139044110040448198997094, −13.84343003088699769136384735717, −13.166949294317977025654156032314, −12.46827745587329221886519479049, −11.37518738689561185872431538738, −10.27957184502059231802241512486, −9.11366306187303044478684570962, −8.089570875693145927933059604, −7.00479411992199808822364606925, −6.34503428814258802424847034356, −5.26594135804838608068421963926, −4.17078926366618894413776171173, −3.28633540975830427411847036103, −2.0724908782559904303032750255, 0.50434027674480173203360270294, 2.26043932526886499400670978703, 3.12410793091927169751323365605, 4.04069048689477916626160111929, 5.47463621269631935905879779354, 5.947941399601461612796030554373, 7.204322041704951783036804236750, 8.50486490822523654875771720811, 9.760350950627988327658285671246, 10.38429085602733817636079016408, 11.35956064949517241173978820728, 12.46177551474681817329670943078, 13.019936483971218269295872328389, 13.85819826897474910430482838802, 14.86682968354460435976429932428, 15.912481506841211219775249483961, 16.301036815975667334542418755709, 18.06860854389576988563904942213, 18.65886798945157976388848776931, 19.54091706307334927727079501406, 20.49538883328187535984646765865, 21.03783020621037751794973084352, 22.1958818370096031252156121768, 22.723927066392813273195863602448, 23.44317741311725284539116157853

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