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Properties

Label 1-3920-3920.309-r0-0-0
Degree $1$
Conductor $3920$
Sign $0.307 - 0.951i$
Analytic cond. $18.2044$
Root an. cond. $18.2044$
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  + (0.974 − 0.222i)3-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (0.433 − 0.900i)13-s + (−0.623 + 0.781i)17-s − i·19-s + (0.623 + 0.781i)23-s + (0.781 − 0.623i)27-s + (−0.781 − 0.623i)29-s + 31-s + (0.222 − 0.974i)33-s + (−0.781 − 0.623i)37-s + (0.222 − 0.974i)39-s + (0.222 + 0.974i)41-s + (0.974 + 0.222i)43-s + ⋯
L(s)  = 1  + (0.974 − 0.222i)3-s + (0.900 − 0.433i)9-s + (0.433 − 0.900i)11-s + (0.433 − 0.900i)13-s + (−0.623 + 0.781i)17-s − i·19-s + (0.623 + 0.781i)23-s + (0.781 − 0.623i)27-s + (−0.781 − 0.623i)29-s + 31-s + (0.222 − 0.974i)33-s + (−0.781 − 0.623i)37-s + (0.222 − 0.974i)39-s + (0.222 + 0.974i)41-s + (0.974 + 0.222i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.307 - 0.951i$
Analytic conductor: \(18.2044\)
Root analytic conductor: \(18.2044\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (309, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (0:\ ),\ 0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.227080000 - 1.620796918i\)
\(L(\frac12)\) \(\approx\) \(2.227080000 - 1.620796918i\)
\(L(1)\) \(\approx\) \(1.536331218 - 0.3991558951i\)
\(L(1)\) \(\approx\) \(1.536331218 - 0.3991558951i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.974 - 0.222i)T \)
11 \( 1 + (0.433 - 0.900i)T \)
13 \( 1 + (0.433 - 0.900i)T \)
17 \( 1 + (-0.623 + 0.781i)T \)
19 \( 1 - iT \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (-0.781 - 0.623i)T \)
31 \( 1 + T \)
37 \( 1 + (-0.781 - 0.623i)T \)
41 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (0.974 + 0.222i)T \)
47 \( 1 + (0.900 + 0.433i)T \)
53 \( 1 + (0.781 - 0.623i)T \)
59 \( 1 + (-0.974 - 0.222i)T \)
61 \( 1 + (-0.781 - 0.623i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.623 - 0.781i)T \)
73 \( 1 + (-0.900 + 0.433i)T \)
79 \( 1 + T \)
83 \( 1 + (-0.433 - 0.900i)T \)
89 \( 1 + (0.900 - 0.433i)T \)
97 \( 1 - T \)
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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

−18.70834884014950894904919312871, −18.2208993524770737502116611891, −17.190728330601747286275942629805, −16.56963324078990985990505567265, −15.81284180288649946093825667101, −15.20101081169486838578183707277, −14.55774589847137050445802476223, −13.88321940330281065282855227533, −13.4513802447045765406324841701, −12.381802581921202583176852573023, −12.0241414310420944951468635, −10.86759530027813871875709948094, −10.334102728700392407676577362476, −9.38664525492094368469390983765, −9.03499104745139534993081364460, −8.345872617234101166650123282613, −7.2955107496213394566136711721, −7.002372712201840500237202010816, −6.02025670275212635647765208507, −4.87040273968754122677077664245, −4.29861834691313747219784655179, −3.65479922636059543734711033489, −2.67445409545097726773539243681, −1.98477263782167206674911064445, −1.20353576252785418193975381164, 0.703407290863366203450102578747, 1.53037971889640210709199082775, 2.53814796413501614293167334963, 3.20095429535863530044866969115, 3.87701693853954857428426378554, 4.6883913242754547413483267327, 5.812482821235757490443339127590, 6.37726930286581068743708913768, 7.32973765970998340293876947475, 7.90811105989070063394845278338, 8.78128214878002532458207799617, 9.03835088920143282115811850174, 9.99118908314708443782958063352, 10.82961019630598768595369216419, 11.389159166023307460029265043674, 12.38720614782657424522669724463, 13.19530341142588503362505231641, 13.462715970874729812090944642730, 14.23013043807780232678815782111, 15.08556594707053489550617325423, 15.48108272847406552141958251289, 16.141546540203506715446306195882, 17.266508136478712574977208638183, 17.6322846863157280538499406947, 18.52873870568307924567178474575

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