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Properties

Label 1-1375-1375.742-r1-0-0
Degree $1$
Conductor $1375$
Sign $-0.918 - 0.395i$
Analytic cond. $147.764$
Root an. cond. $147.764$
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  + (0.904 − 0.425i)2-s + (0.844 − 0.535i)3-s + (0.637 − 0.770i)4-s + (0.535 − 0.844i)6-s + (0.587 − 0.809i)7-s + (0.248 − 0.968i)8-s + (0.425 − 0.904i)9-s + (0.125 − 0.992i)12-s + (0.904 + 0.425i)13-s + (0.187 − 0.982i)14-s + (−0.187 − 0.982i)16-s + (−0.248 + 0.968i)17-s − i·18-s + (−0.0627 − 0.998i)19-s + (0.0627 − 0.998i)21-s + ⋯
L(s)  = 1  + (0.904 − 0.425i)2-s + (0.844 − 0.535i)3-s + (0.637 − 0.770i)4-s + (0.535 − 0.844i)6-s + (0.587 − 0.809i)7-s + (0.248 − 0.968i)8-s + (0.425 − 0.904i)9-s + (0.125 − 0.992i)12-s + (0.904 + 0.425i)13-s + (0.187 − 0.982i)14-s + (−0.187 − 0.982i)16-s + (−0.248 + 0.968i)17-s − i·18-s + (−0.0627 − 0.998i)19-s + (0.0627 − 0.998i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.918 - 0.395i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.918 - 0.395i$
Analytic conductor: \(147.764\)
Root analytic conductor: \(147.764\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (742, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (1:\ ),\ -0.918 - 0.395i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.230470262 - 5.968421245i\)
\(L(\frac12)\) \(\approx\) \(1.230470262 - 5.968421245i\)
\(L(1)\) \(\approx\) \(1.969495072 - 1.711247902i\)
\(L(1)\) \(\approx\) \(1.969495072 - 1.711247902i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.904 - 0.425i)T \)
3 \( 1 + (0.844 - 0.535i)T \)
7 \( 1 + (0.587 - 0.809i)T \)
13 \( 1 + (0.904 + 0.425i)T \)
17 \( 1 + (-0.248 + 0.968i)T \)
19 \( 1 + (-0.0627 - 0.998i)T \)
23 \( 1 + (-0.904 + 0.425i)T \)
29 \( 1 + (-0.535 - 0.844i)T \)
31 \( 1 + (0.535 - 0.844i)T \)
37 \( 1 + (0.481 + 0.876i)T \)
41 \( 1 + (-0.992 - 0.125i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.770 - 0.637i)T \)
53 \( 1 + (0.844 - 0.535i)T \)
59 \( 1 + (0.425 - 0.904i)T \)
61 \( 1 + (-0.187 + 0.982i)T \)
67 \( 1 + (0.770 - 0.637i)T \)
71 \( 1 + (0.0627 - 0.998i)T \)
73 \( 1 + (0.982 + 0.187i)T \)
79 \( 1 + (0.637 - 0.770i)T \)
83 \( 1 + (-0.770 + 0.637i)T \)
89 \( 1 + (0.992 - 0.125i)T \)
97 \( 1 + (0.998 + 0.0627i)T \)
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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

−21.06635885714230792680861984617, −20.43581603322836882421887306603, −19.88781386473319298076636636567, −18.56175923178195836298326425358, −18.11376222690597910208757758663, −16.87761765786045206752287511500, −15.96527906848809776280852665085, −15.70763438149211842420769822623, −14.73589924302529693869009345835, −14.27965885053081703555573661822, −13.559728131703718890858160403870, −12.698563758192609670375927967492, −11.8865290709900813365908880561, −11.042522948447198788595880277496, −10.19868523681382281325359588954, −9.00101421777061536128903816419, −8.369331653914780160800467786212, −7.773656128977976619215010386477, −6.71017753046346965750594754969, −5.64043404227475508898750094371, −5.04857684149258144044479899922, −4.09140597531735233595499462490, −3.3339868739816145301238989253, −2.46930281516275859297061972220, −1.619395194098296737759979705291, 0.656580172770290664371928426372, 1.625309484968959487035935449486, 2.23979232550171417124043535108, 3.50530909666810311928373759849, 3.98622224508041389330836531391, 4.86842596914582939861690257606, 6.20654083314219161338175446658, 6.687826370533627110810571914786, 7.74314355244548843866666159107, 8.407092809482208045475317741998, 9.560289699272650895836368893956, 10.312522424895314736369768634497, 11.34347241901868218617709862965, 11.77985838624108225653126950951, 13.03909680911556272129861138603, 13.445516550299976270758743890161, 13.91964995751741864442377004481, 14.95206252319891560934033244039, 15.26978339368658301023519540856, 16.38182331580719279686633672860, 17.39415577452195234404343917938, 18.29287133359557403268951378939, 19.04127445014321127471771569514, 19.88346273484165637765957426462, 20.215768961641521603822600794173

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