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Properties

Label 1-2664-2664.1285-r0-0-0
Degree $1$
Conductor $2664$
Sign $0.546 - 0.837i$
Analytic cond. $12.3715$
Root an. cond. $12.3715$
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  + (−0.5 − 0.866i)5-s + 7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)35-s + (−0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)5-s + 7-s + (0.5 + 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)29-s + (0.5 − 0.866i)31-s + (−0.5 − 0.866i)35-s + (−0.5 − 0.866i)41-s + (−0.5 − 0.866i)43-s + (−0.5 − 0.866i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2664 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.546 - 0.837i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2664\)    =    \(2^{3} \cdot 3^{2} \cdot 37\)
Sign: $0.546 - 0.837i$
Analytic conductor: \(12.3715\)
Root analytic conductor: \(12.3715\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2664} (1285, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2664,\ (0:\ ),\ 0.546 - 0.837i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.457626077 - 0.7890672285i\)
\(L(\frac12)\) \(\approx\) \(1.457626077 - 0.7890672285i\)
\(L(1)\) \(\approx\) \(1.096215256 - 0.2048777902i\)
\(L(1)\) \(\approx\) \(1.096215256 - 0.2048777902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
37 \( 1 \)
good5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.5 + 0.866i)T \)
59 \( 1 + T \)
61 \( 1 + T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + T \)
79 \( 1 - T \)
83 \( 1 + (0.5 - 0.866i)T \)
89 \( 1 + (0.5 + 0.866i)T \)
97 \( 1 + (0.5 + 0.866i)T \)
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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

−19.47410888950860678803480812826, −18.70618881600275075057151380799, −18.10461891154492282900483525813, −17.375950733247536750922878031799, −16.59260703866024542785703825107, −15.90004442626215544135814718876, −15.006000239770370510026622952, −14.44046236891357522853839966686, −14.01361731538217611430883715027, −13.11883473518666955639697869854, −11.94930353022926085965930956341, −11.39794475517774356209289362264, −11.14711001095838568347539321386, −10.121267793986835122647825221509, −9.23455674083039747910069983434, −8.504401229565311918449444184765, −7.69752038227246417754076241636, −6.9837325366079494344040204405, −6.419175028789577480406622698683, −5.20743579672021014682409048373, −4.656529942758767622531234639170, −3.615281535465638204751105923446, −2.959118168338840350904663359266, −1.97531602723750407206100912627, −0.95686974346214562499056117877, 0.63789797373158624707446976709, 1.632906770945987667450552023835, 2.348602837773391860515740699015, 3.8166861655805678064275745860, 4.21853378266099915180718055178, 5.14409497992975977877157437684, 5.68667862091401648041552327232, 6.87074544755388236563791820820, 7.73949641661833404066876097402, 8.23804258292088408052015560647, 8.85302512066659360341024046431, 9.954330218681768039354052004474, 10.43725514848050182485002444256, 11.53624510712038160315748308148, 12.06732592837531866466821898781, 12.67150712767393149183878138909, 13.35264256230506032151043875955, 14.5199732149413899050537341207, 14.9437296763752736084027116965, 15.46647945664166659805866434616, 16.60268254762613048769456512013, 17.2018075699164555370103525767, 17.45476194295152766220231650454, 18.6370079781438830473018195227, 19.164748995222975205759992482

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