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Properties

Label 1-1375-1375.673-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.952 + 0.306i$
Analytic cond. $6.38547$
Root an. cond. $6.38547$
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  + (0.248 + 0.968i)2-s + (−0.904 − 0.425i)3-s + (−0.876 + 0.481i)4-s + (0.187 − 0.982i)6-s + (0.951 + 0.309i)7-s + (−0.684 − 0.728i)8-s + (0.637 + 0.770i)9-s + (0.998 − 0.0627i)12-s + (−0.770 + 0.637i)13-s + (−0.0627 + 0.998i)14-s + (0.535 − 0.844i)16-s + (0.904 − 0.425i)17-s + (−0.587 + 0.809i)18-s + (−0.187 + 0.982i)19-s + (−0.728 − 0.684i)21-s + ⋯
L(s)  = 1  + (0.248 + 0.968i)2-s + (−0.904 − 0.425i)3-s + (−0.876 + 0.481i)4-s + (0.187 − 0.982i)6-s + (0.951 + 0.309i)7-s + (−0.684 − 0.728i)8-s + (0.637 + 0.770i)9-s + (0.998 − 0.0627i)12-s + (−0.770 + 0.637i)13-s + (−0.0627 + 0.998i)14-s + (0.535 − 0.844i)16-s + (0.904 − 0.425i)17-s + (−0.587 + 0.809i)18-s + (−0.187 + 0.982i)19-s + (−0.728 − 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.952 + 0.306i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (673, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.952 + 0.306i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1377255428 + 0.8785094075i\)
\(L(\frac12)\) \(\approx\) \(0.1377255428 + 0.8785094075i\)
\(L(1)\) \(\approx\) \(0.6861817134 + 0.4666446127i\)
\(L(1)\) \(\approx\) \(0.6861817134 + 0.4666446127i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.248 + 0.968i)T \)
3 \( 1 + (-0.904 - 0.425i)T \)
7 \( 1 + (0.951 + 0.309i)T \)
13 \( 1 + (-0.770 + 0.637i)T \)
17 \( 1 + (0.904 - 0.425i)T \)
19 \( 1 + (-0.187 + 0.982i)T \)
23 \( 1 + (-0.844 + 0.535i)T \)
29 \( 1 + (-0.992 - 0.125i)T \)
31 \( 1 + (0.728 - 0.684i)T \)
37 \( 1 + (-0.770 + 0.637i)T \)
41 \( 1 + (0.637 + 0.770i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (-0.982 + 0.187i)T \)
53 \( 1 + (0.982 - 0.187i)T \)
59 \( 1 + (0.929 - 0.368i)T \)
61 \( 1 + (0.929 + 0.368i)T \)
67 \( 1 + (-0.904 + 0.425i)T \)
71 \( 1 + (-0.425 + 0.904i)T \)
73 \( 1 + (-0.248 - 0.968i)T \)
79 \( 1 + (0.728 + 0.684i)T \)
83 \( 1 + (-0.684 - 0.728i)T \)
89 \( 1 + (-0.0627 + 0.998i)T \)
97 \( 1 + (-0.481 - 0.876i)T \)
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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

−20.760836686332887275372575211703, −19.85773065333172085583568566453, −19.09641391051501719162640305175, −18.04718965338516203894423473497, −17.633671426812311778785937416798, −17.01505646387257075726790142825, −15.95623320691524748176411220440, −14.90280185926041937360321705625, −14.5019189779314422481044416061, −13.44419730629492051181718005504, −12.48178822287943367654278207355, −12.03384110771731048196797783626, −11.16320797122457449318165139520, −10.55170530767783708861210516651, −10.00844533495883791634278873943, −9.04778958194018151356137656220, −8.05695188991905383361799711156, −7.04156693671037316196670797698, −5.7552392987747966072058464845, −5.2195805794634650924344699919, −4.4311097646805705128688013609, −3.71137891064356035983682062972, −2.51467935594529914815573729422, −1.4332353047382318680863093839, −0.40628242231552701607519773677, 1.20139296356001058010286327872, 2.30700618503734289058871303155, 3.87542164619258664791819492574, 4.67140792286558505196921778938, 5.47483289482623073252356955666, 5.97282998159875417888113764980, 7.03868812825267140688105135269, 7.69867334503865294251194908324, 8.279614967953333665122030609064, 9.52460213215900885706609351383, 10.21542603039800403237810293699, 11.59248085945723678807880914074, 11.90035679288037330289132371688, 12.73675498289199538565848207296, 13.684044040954146048332130666947, 14.3962905281097947575833630878, 15.0310375330994497965924330484, 16.09066400255466022565419334274, 16.635574711586532663888247129615, 17.35407300640278982317914234788, 17.925099613329463390271859560587, 18.71251821596027613239404652547, 19.222374941310355396825811896673, 20.82171910154635982712152643469, 21.31880019962452126380458010791

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