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Properties

Label 1-2864-2864.115-r0-0-0
Degree $1$
Conductor $2864$
Sign $0.161 - 0.986i$
Analytic cond. $13.3003$
Root an. cond. $13.3003$
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  + (−0.772 + 0.635i)3-s + (−0.815 + 0.579i)5-s + (−0.0881 + 0.996i)7-s + (0.192 − 0.981i)9-s + (−0.0352 + 0.999i)11-s + (0.835 + 0.550i)13-s + (0.261 − 0.965i)15-s + (−0.825 − 0.564i)17-s + (−0.904 − 0.427i)19-s + (−0.564 − 0.825i)21-s + (0.949 + 0.312i)23-s + (0.329 − 0.944i)25-s + (0.474 + 0.880i)27-s + (0.140 + 0.990i)29-s + (−0.844 − 0.535i)31-s + ⋯
L(s)  = 1  + (−0.772 + 0.635i)3-s + (−0.815 + 0.579i)5-s + (−0.0881 + 0.996i)7-s + (0.192 − 0.981i)9-s + (−0.0352 + 0.999i)11-s + (0.835 + 0.550i)13-s + (0.261 − 0.965i)15-s + (−0.825 − 0.564i)17-s + (−0.904 − 0.427i)19-s + (−0.564 − 0.825i)21-s + (0.949 + 0.312i)23-s + (0.329 − 0.944i)25-s + (0.474 + 0.880i)27-s + (0.140 + 0.990i)29-s + (−0.844 − 0.535i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.161 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2864\)    =    \(2^{4} \cdot 179\)
Sign: $0.161 - 0.986i$
Analytic conductor: \(13.3003\)
Root analytic conductor: \(13.3003\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2864} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2864,\ (0:\ ),\ 0.161 - 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.06599093960 + 0.05608175420i\)
\(L(\frac12)\) \(\approx\) \(-0.06599093960 + 0.05608175420i\)
\(L(1)\) \(\approx\) \(0.5222843358 + 0.2999017170i\)
\(L(1)\) \(\approx\) \(0.5222843358 + 0.2999017170i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
179 \( 1 \)
good3 \( 1 + (-0.772 + 0.635i)T \)
5 \( 1 + (-0.815 + 0.579i)T \)
7 \( 1 + (-0.0881 + 0.996i)T \)
11 \( 1 + (-0.0352 + 0.999i)T \)
13 \( 1 + (0.835 + 0.550i)T \)
17 \( 1 + (-0.825 - 0.564i)T \)
19 \( 1 + (-0.904 - 0.427i)T \)
23 \( 1 + (0.949 + 0.312i)T \)
29 \( 1 + (0.140 + 0.990i)T \)
31 \( 1 + (-0.844 - 0.535i)T \)
37 \( 1 + (-0.140 + 0.990i)T \)
41 \( 1 + (0.158 + 0.987i)T \)
43 \( 1 + (0.944 - 0.329i)T \)
47 \( 1 + (-0.227 - 0.973i)T \)
53 \( 1 + (-0.474 + 0.880i)T \)
59 \( 1 + (-0.621 - 0.783i)T \)
61 \( 1 + (-0.442 - 0.896i)T \)
67 \( 1 + (-0.593 - 0.804i)T \)
71 \( 1 + (-0.997 + 0.0705i)T \)
73 \( 1 + (0.760 + 0.648i)T \)
79 \( 1 + (-0.688 + 0.725i)T \)
83 \( 1 + (-0.932 - 0.362i)T \)
89 \( 1 + (-0.977 + 0.210i)T \)
97 \( 1 + (0.0529 - 0.998i)T \)
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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

−18.7684015614782174478917851228, −17.7160609673047948290478242046, −17.274816173768967400963402167625, −16.427127835246018215746481717183, −16.1493119767764127262830307575, −15.2700458154414663387207495743, −14.246482508477177032058595745120, −13.36880473227279620768815108046, −12.95866159434817715626764802461, −12.37176997652899775796754491882, −11.30354945705228700014976953729, −10.89092265979496566972322179945, −10.48001543279386319163125219257, −9.0065031544210009012368287511, −8.392505270089406831454863452818, −7.709548223448390608413734271690, −7.00809740187564270273180041463, −6.15014050993025954374310062832, −5.56654400373476096957046629056, −4.42637032347647300447512725167, −4.00167305372430223532407657897, −2.97135772501848074423920977741, −1.61657224005999285987519033718, −0.85119295696568826899026299907, −0.03920586303324208444910221953, 1.53744783017372560882477391214, 2.64122895765768987609496251351, 3.44161293125429149244358793983, 4.41221722895692842678530936041, 4.82642592918918464580570467940, 5.860770177102582507895979917750, 6.69789486560115073540458321430, 7.06256316651424723220838979190, 8.31208172907515683193125305382, 9.08327196987292609726693101095, 9.59317395035416066387520822875, 10.76906750322444939252468777400, 11.08808032732151902819246174122, 11.76739838088922464681521769780, 12.45502458814187816746400086131, 13.122017603988811889979084430762, 14.37283984836319662393717829267, 15.12824261590057782909487652284, 15.4387478635385505775938659827, 16.01948987246200154168784929813, 16.85959142633294504549211017358, 17.64285969955467936151496840661, 18.483314011710900401002684260, 18.63306798773518510844409013396, 19.786306026303074088068277370627

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