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Properties

Label 1-1375-1375.688-r0-0-0
Degree $1$
Conductor $1375$
Sign $-0.950 + 0.312i$
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  + (0.770 − 0.637i)2-s + (0.684 + 0.728i)3-s + (0.187 − 0.982i)4-s + (0.992 + 0.125i)6-s + (−0.951 + 0.309i)7-s + (−0.481 − 0.876i)8-s + (−0.0627 + 0.998i)9-s + (0.844 − 0.535i)12-s + (−0.998 − 0.0627i)13-s + (−0.535 + 0.844i)14-s + (−0.929 − 0.368i)16-s + (−0.684 + 0.728i)17-s + (0.587 + 0.809i)18-s + (−0.992 − 0.125i)19-s + (−0.876 − 0.481i)21-s + ⋯
L(s)  = 1  + (0.770 − 0.637i)2-s + (0.684 + 0.728i)3-s + (0.187 − 0.982i)4-s + (0.992 + 0.125i)6-s + (−0.951 + 0.309i)7-s + (−0.481 − 0.876i)8-s + (−0.0627 + 0.998i)9-s + (0.844 − 0.535i)12-s + (−0.998 − 0.0627i)13-s + (−0.535 + 0.844i)14-s + (−0.929 − 0.368i)16-s + (−0.684 + 0.728i)17-s + (0.587 + 0.809i)18-s + (−0.992 − 0.125i)19-s + (−0.876 − 0.481i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1375 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.950 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1375\)    =    \(5^{3} \cdot 11\)
Sign: $-0.950 + 0.312i$
Analytic conductor: \(6.38547\)
Root analytic conductor: \(6.38547\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1375} (688, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1375,\ (0:\ ),\ -0.950 + 0.312i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01036873950 - 0.06480549793i\)
\(L(\frac12)\) \(\approx\) \(0.01036873950 - 0.06480549793i\)
\(L(1)\) \(\approx\) \(1.190853527 - 0.2015760833i\)
\(L(1)\) \(\approx\) \(1.190853527 - 0.2015760833i\)

Euler product

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

−21.40423343331072398208391257877, −20.38617369398087330861281325699, −19.838738762849911248920840587013, −19.13139836306648230077803890927, −18.17219437462855181688652277320, −17.314267353139418249933215518025, −16.72970235730419248097346836536, −15.626482534140190500274355208970, −15.22825729461040438380577588101, −14.15627543324835376488296368206, −13.74763845279142127815144262691, −12.943853811529383211301046751032, −12.36966422830367760042051398430, −11.67178126561586053986800443320, −10.36144791769745492199180128856, −9.29977562291323695074530277761, −8.65926151803978043642764584606, −7.63052730379890795835973215877, −6.98186558726314547741004555939, −6.508532679368932762865172230448, −5.4560375794896168788932538501, −4.39461644702676988283804883602, −3.475737057535903674808252537364, −2.77842361139679753886810275819, −1.85619163986227683005788770566, 0.01484068929116645730784387536, 2.00494335774436338409792534480, 2.553369508137557359122209146398, 3.428771746296442571845037122083, 4.28415531297742279074797687336, 4.89005803604227927951729911420, 6.05374979894848675775495481528, 6.69459768806220644021195181538, 8.03384211216789207259082807975, 8.971604545332374050686690286350, 9.75061028500129665690434324488, 10.27533636051825682484900927702, 11.05529362712900019636375062870, 12.12412873865857773933454499458, 12.82959084712265712915548227402, 13.472487270579985356646990722485, 14.31549362389469596084638755712, 15.20622512117148712742328585962, 15.37999552189376921094458211932, 16.44510608107264933275371323103, 17.22690388771319112090351472543, 18.62199837919535122542155544073, 19.29285196907326461016642850709, 19.718229229087140333237267752030, 20.44630627149518026598452909947

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