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Properties

Label 1-39e2-1521.76-r1-0-0
Degree $1$
Conductor $1521$
Sign $0.963 + 0.267i$
Analytic cond. $163.454$
Root an. cond. $163.454$
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  + (0.0804 − 0.996i)2-s + (−0.987 − 0.160i)4-s + (−0.979 − 0.200i)5-s + (−0.239 + 0.970i)7-s + (−0.239 + 0.970i)8-s + (−0.278 + 0.960i)10-s + (−0.903 − 0.428i)11-s + (0.948 + 0.316i)14-s + (0.948 + 0.316i)16-s + (−0.278 − 0.960i)17-s + (0.866 − 0.5i)19-s + (0.935 + 0.354i)20-s + (−0.5 + 0.866i)22-s − 23-s + (0.919 + 0.391i)25-s + ⋯
L(s)  = 1  + (0.0804 − 0.996i)2-s + (−0.987 − 0.160i)4-s + (−0.979 − 0.200i)5-s + (−0.239 + 0.970i)7-s + (−0.239 + 0.970i)8-s + (−0.278 + 0.960i)10-s + (−0.903 − 0.428i)11-s + (0.948 + 0.316i)14-s + (0.948 + 0.316i)16-s + (−0.278 − 0.960i)17-s + (0.866 − 0.5i)19-s + (0.935 + 0.354i)20-s + (−0.5 + 0.866i)22-s − 23-s + (0.919 + 0.391i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.963 + 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $0.963 + 0.267i$
Analytic conductor: \(163.454\)
Root analytic conductor: \(163.454\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (76, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1521,\ (1:\ ),\ 0.963 + 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1888296328 + 0.02570355077i\)
\(L(\frac12)\) \(\approx\) \(0.1888296328 + 0.02570355077i\)
\(L(1)\) \(\approx\) \(0.5240457999 - 0.3319006152i\)
\(L(1)\) \(\approx\) \(0.5240457999 - 0.3319006152i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.0804 - 0.996i)T \)
5 \( 1 + (-0.979 - 0.200i)T \)
7 \( 1 + (-0.239 + 0.970i)T \)
11 \( 1 + (-0.903 - 0.428i)T \)
17 \( 1 + (-0.278 - 0.960i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.996 - 0.0804i)T \)
31 \( 1 + (-0.391 - 0.919i)T \)
37 \( 1 + (0.600 - 0.799i)T \)
41 \( 1 + (0.464 + 0.885i)T \)
43 \( 1 + (-0.120 - 0.992i)T \)
47 \( 1 + (-0.774 + 0.632i)T \)
53 \( 1 + (-0.970 - 0.239i)T \)
59 \( 1 + (-0.316 - 0.948i)T \)
61 \( 1 + (-0.970 + 0.239i)T \)
67 \( 1 + (-0.935 - 0.354i)T \)
71 \( 1 + (0.534 - 0.845i)T \)
73 \( 1 + (0.822 - 0.568i)T \)
79 \( 1 + (-0.632 - 0.774i)T \)
83 \( 1 + (-0.534 - 0.845i)T \)
89 \( 1 + (0.866 + 0.5i)T \)
97 \( 1 + (-0.663 - 0.748i)T \)
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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

−20.16888454283735893456990097772, −19.67959394948398940855838149902, −18.64817078609003891400262757217, −18.12415327014284212903385514932, −17.24534358375721664475143344175, −16.40808210298245466017140181633, −15.952529785966498972959519641798, −15.183086410781217171938811194617, −14.4959879743452183320129801432, −13.70489591649568340728237689677, −12.87977687456217439350916111405, −12.28629909995357188245722274554, −11.10255666148384171945026782559, −10.30520723097116464151668615600, −9.60031028930398893154778670441, −8.38951230602361771443313743979, −7.793825039485897735304803527, −7.27891899164639540063160053020, −6.44333726752157381247537288588, −5.46613630630712865763433007657, −4.48507401369151456408641262044, −3.86966872879781011041993932686, −3.07759031949922856340070128590, −1.364564769521955888294232827581, −0.07263263214824472124365353352, 0.48314659683463448386422525015, 1.93100150416540451343516532203, 2.8426357515904176257992379475, 3.45366913869050763552142078129, 4.54353537786198707413934946313, 5.24114697163088160196033897196, 6.06956678577401313994917587938, 7.59120404433751863017397338122, 8.071781855428020986250144262692, 9.17125942825288332764704641944, 9.50382551651482166796173753448, 10.756510575683166127190355768, 11.42600214017385338346833287271, 11.90873170628306055438192935970, 12.76642343948026354188312901717, 13.3362131816988094482940270452, 14.304007197782017719939325674729, 15.22081716913176369235080467963, 15.88057626645017538670240590512, 16.55113357906226322314736610209, 17.91488326110685902434405942302, 18.393883567492829554020479634126, 18.95086861034137435363109477450, 19.78874030156736809647841505943, 20.37584716424909429643138091280

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