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Properties

Label 1-3328-3328.163-r0-0-0
Degree $1$
Conductor $3328$
Sign $0.986 + 0.164i$
Analytic cond. $15.4551$
Root an. cond. $15.4551$
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  + (−0.412 − 0.910i)3-s + (0.881 − 0.471i)5-s + (−0.321 + 0.946i)7-s + (−0.659 + 0.751i)9-s + (0.986 + 0.162i)11-s + (−0.793 − 0.608i)15-s + (−0.608 − 0.793i)17-s + (−0.973 − 0.227i)19-s + (0.995 − 0.0980i)21-s + (0.896 + 0.442i)23-s + (0.555 − 0.831i)25-s + (0.956 + 0.290i)27-s + (−0.352 + 0.935i)29-s + (0.707 + 0.707i)31-s + (−0.258 − 0.965i)33-s + ⋯
L(s)  = 1  + (−0.412 − 0.910i)3-s + (0.881 − 0.471i)5-s + (−0.321 + 0.946i)7-s + (−0.659 + 0.751i)9-s + (0.986 + 0.162i)11-s + (−0.793 − 0.608i)15-s + (−0.608 − 0.793i)17-s + (−0.973 − 0.227i)19-s + (0.995 − 0.0980i)21-s + (0.896 + 0.442i)23-s + (0.555 − 0.831i)25-s + (0.956 + 0.290i)27-s + (−0.352 + 0.935i)29-s + (0.707 + 0.707i)31-s + (−0.258 − 0.965i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.164i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3328\)    =    \(2^{8} \cdot 13\)
Sign: $0.986 + 0.164i$
Analytic conductor: \(15.4551\)
Root analytic conductor: \(15.4551\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3328} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3328,\ (0:\ ),\ 0.986 + 0.164i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.528212353 + 0.1262650959i\)
\(L(\frac12)\) \(\approx\) \(1.528212353 + 0.1262650959i\)
\(L(1)\) \(\approx\) \(1.040491404 - 0.1590216901i\)
\(L(1)\) \(\approx\) \(1.040491404 - 0.1590216901i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
13 \( 1 \)
good3 \( 1 + (-0.412 - 0.910i)T \)
5 \( 1 + (0.881 - 0.471i)T \)
7 \( 1 + (-0.321 + 0.946i)T \)
11 \( 1 + (0.986 + 0.162i)T \)
17 \( 1 + (-0.608 - 0.793i)T \)
19 \( 1 + (-0.973 - 0.227i)T \)
23 \( 1 + (0.896 + 0.442i)T \)
29 \( 1 + (-0.352 + 0.935i)T \)
31 \( 1 + (0.707 + 0.707i)T \)
37 \( 1 + (0.227 + 0.973i)T \)
41 \( 1 + (-0.896 - 0.442i)T \)
43 \( 1 + (-0.412 + 0.910i)T \)
47 \( 1 + (0.382 - 0.923i)T \)
53 \( 1 + (-0.634 + 0.773i)T \)
59 \( 1 + (0.999 + 0.0327i)T \)
61 \( 1 + (0.812 + 0.582i)T \)
67 \( 1 + (-0.412 - 0.910i)T \)
71 \( 1 + (-0.946 - 0.321i)T \)
73 \( 1 + (0.980 + 0.195i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.956 - 0.290i)T \)
89 \( 1 + (0.997 + 0.0654i)T \)
97 \( 1 + (-0.965 + 0.258i)T \)
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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

−19.041806911017283169537585000627, −17.800124528066813378275951133031, −17.16943509749774187347231292140, −17.03062270380146192839348651106, −16.24851379862140242666071310348, −15.229874627382493027414132569977, −14.73440015794936528105574049026, −14.11278048933095317597878032380, −13.28950698658939550933816942292, −12.68814583211436991773453610821, −11.51047506466905530459686361249, −11.022257223162415425936880756432, −10.31217276802660734726355723360, −9.86157867945716059690572218393, −9.09277872681249030931875777488, −8.41332984471995768013388034653, −7.151480546447383832029559674741, −6.33637953742769519171777759848, −6.16227655066358300217194156367, −5.030346047945120408993032293358, −4.13945231954210611214638706715, −3.73405114060750267074238071515, −2.70667368869834213095147221359, −1.71247880819944921684206945899, −0.55175492438296885817536999500, 0.94332191019304226518162831088, 1.74069054442645455445914656290, 2.41573733647469213010921832305, 3.2527977399423272028104154078, 4.74036209045722661267497239025, 5.153091636816855799081231043478, 6.108061910724595990668326809516, 6.575661206910691471237163577758, 7.16667799942179705989436048570, 8.50810859192860886618236729912, 8.81674989974992758816800465567, 9.538008270388266234324826592002, 10.46022494692855059912409042235, 11.40844539933463086889679192834, 11.948732938443188392223447828286, 12.64870501284585917189711288763, 13.20595229226923141200421512754, 13.80506060283335868181513445260, 14.63463124074486511380653913886, 15.36866348110185619801714039165, 16.38131994864452084901158670743, 16.9050283045007522569033295113, 17.564423111821636647921412559144, 18.09300881563772711575309192860, 18.807880208792330846749069960815

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