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Properties

Label 1-2668-2668.403-r0-0-0
Degree $1$
Conductor $2668$
Sign $-0.424 - 0.905i$
Analytic cond. $12.3901$
Root an. cond. $12.3901$
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  + (−0.925 + 0.377i)3-s + (0.523 − 0.852i)5-s + (0.862 − 0.505i)7-s + (0.714 − 0.699i)9-s + (−0.607 − 0.794i)11-s + (0.933 − 0.359i)13-s + (−0.162 + 0.986i)15-s + (−0.755 − 0.654i)17-s + (0.0407 + 0.999i)19-s + (−0.607 + 0.794i)21-s + (−0.452 − 0.891i)25-s + (−0.396 + 0.917i)27-s + (−0.670 − 0.742i)31-s + (0.862 + 0.505i)33-s + (0.0203 − 0.999i)35-s + ⋯
L(s)  = 1  + (−0.925 + 0.377i)3-s + (0.523 − 0.852i)5-s + (0.862 − 0.505i)7-s + (0.714 − 0.699i)9-s + (−0.607 − 0.794i)11-s + (0.933 − 0.359i)13-s + (−0.162 + 0.986i)15-s + (−0.755 − 0.654i)17-s + (0.0407 + 0.999i)19-s + (−0.607 + 0.794i)21-s + (−0.452 − 0.891i)25-s + (−0.396 + 0.917i)27-s + (−0.670 − 0.742i)31-s + (0.862 + 0.505i)33-s + (0.0203 − 0.999i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2668 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.424 - 0.905i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2668\)    =    \(2^{2} \cdot 23 \cdot 29\)
Sign: $-0.424 - 0.905i$
Analytic conductor: \(12.3901\)
Root analytic conductor: \(12.3901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2668} (403, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2668,\ (0:\ ),\ -0.424 - 0.905i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6869029935 - 1.080417266i\)
\(L(\frac12)\) \(\approx\) \(0.6869029935 - 1.080417266i\)
\(L(1)\) \(\approx\) \(0.8996719391 - 0.2913581838i\)
\(L(1)\) \(\approx\) \(0.8996719391 - 0.2913581838i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 \)
29 \( 1 \)
good3 \( 1 + (-0.925 + 0.377i)T \)
5 \( 1 + (0.523 - 0.852i)T \)
7 \( 1 + (0.862 - 0.505i)T \)
11 \( 1 + (-0.607 - 0.794i)T \)
13 \( 1 + (0.933 - 0.359i)T \)
17 \( 1 + (-0.755 - 0.654i)T \)
19 \( 1 + (0.0407 + 0.999i)T \)
31 \( 1 + (-0.670 - 0.742i)T \)
37 \( 1 + (0.699 + 0.714i)T \)
41 \( 1 + (-0.540 - 0.841i)T \)
43 \( 1 + (0.670 - 0.742i)T \)
47 \( 1 + (0.974 - 0.222i)T \)
53 \( 1 + (-0.301 - 0.953i)T \)
59 \( 1 + (0.142 + 0.989i)T \)
61 \( 1 + (0.574 - 0.818i)T \)
67 \( 1 + (0.794 + 0.607i)T \)
71 \( 1 + (-0.979 - 0.202i)T \)
73 \( 1 + (0.965 - 0.262i)T \)
79 \( 1 + (0.0815 - 0.996i)T \)
83 \( 1 + (-0.947 + 0.320i)T \)
89 \( 1 + (0.162 + 0.986i)T \)
97 \( 1 + (-0.994 - 0.101i)T \)
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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

−19.33050326315756884689770235698, −18.46111607935563063671378543123, −18.10448722729725743602219109601, −17.6428914741306424143872503566, −16.99969103894611032654429706112, −15.85250027983264737224394812659, −15.41683181379711641500940510434, −14.587949556076991821234005877541, −13.79403596143286665156344446893, −13.034733355524215783717566037959, −12.46925500466571615203019153663, −11.29543376565158072509290422855, −11.13864323224959886870033519370, −10.457291880579379556425642842721, −9.5157640902365917185062864312, −8.63453641832181480065119350579, −7.67217553227533519226204745673, −7.01044147726617665482035429094, −6.284869281175173770804534064562, −5.63301184764260490168118794343, −4.84504327857323050150450464916, −4.09132212851968120656785426914, −2.684509338568189526898458269986, −2.02259085667374781988620190358, −1.279068566638877589404919506532, 0.497662774723075983724599113292, 1.23254755331165254056621985793, 2.208228397766283713136014538615, 3.6483836954469390338536700286, 4.27575759809922544989023129357, 5.196281690876814942675107388577, 5.59780446286388150498819222212, 6.3442455880891891124216391497, 7.4121965018026220600155726509, 8.27845817610738364839224711164, 8.898933434969553107831321250917, 9.8514006971747147482349821859, 10.56084797679268164346081349820, 11.13608013411561134870660759193, 11.78871759301881187395749453836, 12.67195382005144849272103471341, 13.421350657634719986418059658734, 13.885409545752869161136793969488, 14.96585068934327896462014040525, 15.83337629751558099072386642727, 16.312723119632690721277131656228, 16.97719090716410681483269684662, 17.57199512624245847700520697197, 18.27034039213193210703901621730, 18.747265381425768378343118699776

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