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Properties

Label 1-1375-1375.14-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.307 - 0.951i$
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  + (−0.535 − 0.844i)2-s + (−0.876 + 0.481i)3-s + (−0.425 + 0.904i)4-s + (0.876 + 0.481i)6-s + (−0.309 − 0.951i)7-s + (0.992 − 0.125i)8-s + (0.535 − 0.844i)9-s + (−0.0627 − 0.998i)12-s + (−0.535 + 0.844i)13-s + (−0.637 + 0.770i)14-s + (−0.637 − 0.770i)16-s + (0.992 − 0.125i)17-s − 18-s + (0.728 − 0.684i)19-s + (0.728 + 0.684i)21-s + ⋯
L(s)  = 1  + (−0.535 − 0.844i)2-s + (−0.876 + 0.481i)3-s + (−0.425 + 0.904i)4-s + (0.876 + 0.481i)6-s + (−0.309 − 0.951i)7-s + (0.992 − 0.125i)8-s + (0.535 − 0.844i)9-s + (−0.0627 − 0.998i)12-s + (−0.535 + 0.844i)13-s + (−0.637 + 0.770i)14-s + (−0.637 − 0.770i)16-s + (0.992 − 0.125i)17-s − 18-s + (0.728 − 0.684i)19-s + (0.728 + 0.684i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.307 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.307 - 0.951i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.307 - 0.951i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3806450911 - 0.5231651812i\)
\(L(\frac12)\) \(\approx\) \(0.3806450911 - 0.5231651812i\)
\(L(1)\) \(\approx\) \(0.5487607939 - 0.2237385346i\)
\(L(1)\) \(\approx\) \(0.5487607939 - 0.2237385346i\)

Euler product

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

−21.23449052320501869478580094538, −20.00413455226747563182623931381, −19.10239847631580670708802927385, −18.761937659242911733049529186805, −17.80042201598382926817895877420, −17.49182405709613988227267973916, −16.50227930628828843877004704989, −15.89546311063624411928993366563, −15.27905506974059363989405196346, −14.28172008603668403451380794859, −13.51481779237452808599882845564, −12.40596740643955038439062925978, −12.05233938795455015295752487753, −10.89794319463801818286202353400, −10.01714961951872512848349498957, −9.53427467641792708120441457351, −8.205425142758748483419216968063, −7.79451853931170293951471079998, −6.8352156097555002444280590879, −5.94610827175949916120023718096, −5.50684023232870733623762884076, −4.76744442793227382492037107431, −3.23973911211449981388103587462, −1.92459085826844433902308854291, −0.92976398311289200117153999882, 0.46853011832751452110765327649, 1.36004257165457567093226983334, 2.74034977271836748764820358682, 3.70797397712557982582457430642, 4.444012349151972612964856720978, 5.17989617692746694115404504425, 6.592281832582903789161534721456, 7.14642633903341442170525728852, 8.19361955151886832602881289200, 9.28803283437238183029028271463, 10.00739611348738752231213944300, 10.37271080705132432261835362256, 11.33425842638191439152101319960, 11.990360380180111030632508943601, 12.555070600542597044675879865344, 13.686915650625065048553469680885, 14.25127710595409791632086021044, 15.66502305153874938926529646658, 16.35905863953491034839952201242, 16.95727972413201954543889740776, 17.5047386333857077392402870055, 18.3488653404975424108748725352, 19.11944469286205901446580480288, 19.90221589266306387710689426860, 20.64812720703533198128702743338

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