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Properties

Label 1-2667-2667.839-r1-0-0
Degree $1$
Conductor $2667$
Sign $-0.664 - 0.746i$
Analytic cond. $286.608$
Root an. cond. $286.608$
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  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.623 + 0.781i)5-s + (0.222 + 0.974i)8-s + (−0.900 + 0.433i)10-s + (−0.0747 + 0.997i)11-s + (0.988 + 0.149i)13-s + (−0.222 + 0.974i)16-s + (0.365 + 0.930i)17-s − 19-s − 20-s + (−0.5 + 0.866i)22-s + (0.0747 + 0.997i)23-s + (−0.222 − 0.974i)25-s + (0.826 + 0.563i)26-s + ⋯
L(s)  = 1  + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (−0.623 + 0.781i)5-s + (0.222 + 0.974i)8-s + (−0.900 + 0.433i)10-s + (−0.0747 + 0.997i)11-s + (0.988 + 0.149i)13-s + (−0.222 + 0.974i)16-s + (0.365 + 0.930i)17-s − 19-s − 20-s + (−0.5 + 0.866i)22-s + (0.0747 + 0.997i)23-s + (−0.222 − 0.974i)25-s + (0.826 + 0.563i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2667 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.664 - 0.746i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2667\)    =    \(3 \cdot 7 \cdot 127\)
Sign: $-0.664 - 0.746i$
Analytic conductor: \(286.608\)
Root analytic conductor: \(286.608\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2667} (839, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2667,\ (1:\ ),\ -0.664 - 0.746i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.161012617 + 2.588141026i\)
\(L(\frac12)\) \(\approx\) \(-1.161012617 + 2.588141026i\)
\(L(1)\) \(\approx\) \(1.135086857 + 1.100427735i\)
\(L(1)\) \(\approx\) \(1.135086857 + 1.100427735i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.900 + 0.433i)T \)
5 \( 1 + (-0.623 + 0.781i)T \)
11 \( 1 + (-0.0747 + 0.997i)T \)
13 \( 1 + (0.988 + 0.149i)T \)
17 \( 1 + (0.365 + 0.930i)T \)
19 \( 1 - T \)
23 \( 1 + (0.0747 + 0.997i)T \)
29 \( 1 + (-0.733 - 0.680i)T \)
31 \( 1 + (-0.365 + 0.930i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 + (0.365 + 0.930i)T \)
43 \( 1 + (0.733 + 0.680i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + (0.826 - 0.563i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.222 - 0.974i)T \)
67 \( 1 + (-0.955 + 0.294i)T \)
71 \( 1 + (-0.0747 - 0.997i)T \)
73 \( 1 + (0.900 - 0.433i)T \)
79 \( 1 + (0.0747 + 0.997i)T \)
83 \( 1 + (-0.826 + 0.563i)T \)
89 \( 1 + (0.900 + 0.433i)T \)
97 \( 1 + (-0.733 + 0.680i)T \)
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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

−18.850400545314837406494078729231, −18.34068660404062688641686056419, −16.95991428496385079005875128376, −16.35317259526610338083489181859, −15.87814215516821531485839264228, −15.10490020390362369504057695558, −14.32598062789689723218461645639, −13.539870505413563601979896034792, −13.010154138866818813086950489, −12.2934652898034741905242487754, −11.62544709883809934158842470535, −10.89903726052557628466997805118, −10.41569369243233632458808601050, −9.04439161080490630619234448827, −8.7393624779888072851124578756, −7.62139474485744532730017789883, −6.86614842227660962661739756817, −5.67586487404169799899457059640, −5.527830675630913616588040050712, −4.247054723123045376440868915567, −3.908862561651514748211444142591, −2.9838789739002315236300476338, −2.03197783792283043932891403027, −0.89326173484902636604578896254, −0.360963439555341217912148030425, 1.49982588887060474673583065957, 2.35003120061789537420459606919, 3.37757968408428241633447856010, 3.90903908953547953855363670133, 4.613129387436756748096751327114, 5.668082681644721199046117498834, 6.40962184269330079387903667558, 6.97610223253577763436325252382, 7.85151026002574676288936736456, 8.29755523435171836001472395353, 9.46765475592875778760162049074, 10.566618597504079679384553642431, 11.045620250294581909277737715010, 11.82228312468130505770506978889, 12.56280786492824958983313040775, 13.16720572614154562926734998688, 14.022781288882491735563627063837, 14.77222175729705868362947633335, 15.21777359575481289454807399315, 15.78078611848444438689785164052, 16.59922832099896268331394123452, 17.4386296680995888017922657866, 18.00710867280165828550196842277, 19.02561431129574787113987055521, 19.57984683295993011657219413125

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