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Properties

Label 1-4864-4864.2173-r0-0-0
Degree $1$
Conductor $4864$
Sign $0.900 + 0.434i$
Analytic cond. $22.5883$
Root an. cond. $22.5883$
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  + (0.986 − 0.162i)3-s + (−0.729 + 0.683i)5-s + (−0.980 + 0.195i)7-s + (0.946 − 0.321i)9-s + (0.995 − 0.0980i)11-s + (0.683 − 0.729i)13-s + (−0.608 + 0.793i)15-s + (−0.608 − 0.793i)17-s + (−0.935 + 0.352i)21-s + (−0.442 + 0.896i)23-s + (0.0654 − 0.997i)25-s + (0.881 − 0.471i)27-s + (−0.910 + 0.412i)29-s + (−0.707 + 0.707i)31-s + (0.965 − 0.258i)33-s + ⋯
L(s)  = 1  + (0.986 − 0.162i)3-s + (−0.729 + 0.683i)5-s + (−0.980 + 0.195i)7-s + (0.946 − 0.321i)9-s + (0.995 − 0.0980i)11-s + (0.683 − 0.729i)13-s + (−0.608 + 0.793i)15-s + (−0.608 − 0.793i)17-s + (−0.935 + 0.352i)21-s + (−0.442 + 0.896i)23-s + (0.0654 − 0.997i)25-s + (0.881 − 0.471i)27-s + (−0.910 + 0.412i)29-s + (−0.707 + 0.707i)31-s + (0.965 − 0.258i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 + 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4864\)    =    \(2^{8} \cdot 19\)
Sign: $0.900 + 0.434i$
Analytic conductor: \(22.5883\)
Root analytic conductor: \(22.5883\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4864} (2173, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4864,\ (0:\ ),\ 0.900 + 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.002386074 + 0.4581140178i\)
\(L(\frac12)\) \(\approx\) \(2.002386074 + 0.4581140178i\)
\(L(1)\) \(\approx\) \(1.271576808 + 0.08639235920i\)
\(L(1)\) \(\approx\) \(1.271576808 + 0.08639235920i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 \)
good3 \( 1 + (0.986 - 0.162i)T \)
5 \( 1 + (-0.729 + 0.683i)T \)
7 \( 1 + (-0.980 + 0.195i)T \)
11 \( 1 + (0.995 - 0.0980i)T \)
13 \( 1 + (0.683 - 0.729i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
23 \( 1 + (-0.442 + 0.896i)T \)
29 \( 1 + (-0.910 + 0.412i)T \)
31 \( 1 + (-0.707 + 0.707i)T \)
37 \( 1 + (0.471 + 0.881i)T \)
41 \( 1 + (0.0654 + 0.997i)T \)
43 \( 1 + (0.352 + 0.935i)T \)
47 \( 1 + (-0.130 - 0.991i)T \)
53 \( 1 + (0.812 - 0.582i)T \)
59 \( 1 + (0.683 + 0.729i)T \)
61 \( 1 + (-0.935 - 0.352i)T \)
67 \( 1 + (0.935 + 0.352i)T \)
71 \( 1 + (-0.946 - 0.321i)T \)
73 \( 1 + (0.659 - 0.751i)T \)
79 \( 1 + (0.130 - 0.991i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (0.997 + 0.0654i)T \)
97 \( 1 + (0.258 + 0.965i)T \)
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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

−18.31537910166282498564227052119, −17.05915524286340180972136102347, −16.67686370897257038539153828312, −15.94437180240342739984577690461, −15.4986041969095994272693312970, −14.72766034693008346370834709788, −14.06668741718354543981749358761, −13.334258107973100309801954797395, −12.74431632196424620970809222400, −12.237911103087283810515801502008, −11.26732398599427062850962700856, −10.61392137576451834609020531950, −9.60989543187218809023112140380, −9.09268040088030512327308795239, −8.72243383944030410213138906564, −7.867156428088226886090673957833, −7.131727784360780251238749125303, −6.47800692842276051131932790882, −5.63756819877427733635425279964, −4.28142415246816008195135652793, −4.008416572282732075763987882212, −3.610157883835680968250304860095, −2.39883672318671868945508744860, −1.69293478224091394171245302385, −0.637913311971916523077848100717, 0.78306191928296136581908067263, 1.84044299277430559108236956339, 2.7983776487502804658717568494, 3.45663913234715887384385742262, 3.7073283041030663871497970729, 4.72121151680654313182492875942, 5.955760591007369262884686227298, 6.59574964240853867303725426861, 7.19824153249119152321265796575, 7.8357199024209949642869248766, 8.67456523419725149198170914029, 9.23347896405098017547774513979, 9.88235424472913158047328985367, 10.66293432187606330156998766337, 11.52785184454267352080148247380, 12.05325400209056321725376311440, 13.03899793950390441560444120509, 13.38723214007437653078352115958, 14.19851438941256867501353369376, 14.943179381844982984939776911481, 15.29725996552459897440069197073, 16.13366628434460156552449784484, 16.45151980522949801619882827842, 17.85994855793208626747872264012, 18.21050499706319053378993908924

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