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Properties

Label 1-3968-3968.1243-r0-0-0
Degree $1$
Conductor $3968$
Sign $0.213 + 0.976i$
Analytic cond. $18.4273$
Root an. cond. $18.4273$
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  + (0.716 − 0.697i)3-s + (−0.946 + 0.321i)5-s + (0.0261 + 0.999i)7-s + (0.0261 − 0.999i)9-s + (−0.465 − 0.884i)11-s + (0.859 + 0.511i)13-s + (−0.453 + 0.891i)15-s + (0.777 + 0.629i)17-s + (−0.801 − 0.598i)19-s + (0.716 + 0.697i)21-s + (−0.522 + 0.852i)23-s + (0.793 − 0.608i)25-s + (−0.678 − 0.734i)27-s + (−0.785 + 0.619i)29-s + (−0.951 − 0.309i)33-s + ⋯
L(s)  = 1  + (0.716 − 0.697i)3-s + (−0.946 + 0.321i)5-s + (0.0261 + 0.999i)7-s + (0.0261 − 0.999i)9-s + (−0.465 − 0.884i)11-s + (0.859 + 0.511i)13-s + (−0.453 + 0.891i)15-s + (0.777 + 0.629i)17-s + (−0.801 − 0.598i)19-s + (0.716 + 0.697i)21-s + (−0.522 + 0.852i)23-s + (0.793 − 0.608i)25-s + (−0.678 − 0.734i)27-s + (−0.785 + 0.619i)29-s + (−0.951 − 0.309i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.213 + 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $0.213 + 0.976i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ 0.213 + 0.976i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9397325063 + 0.7561629070i\)
\(L(\frac12)\) \(\approx\) \(0.9397325063 + 0.7561629070i\)
\(L(1)\) \(\approx\) \(1.041249206 + 0.002044263084i\)
\(L(1)\) \(\approx\) \(1.041249206 + 0.002044263084i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.716 - 0.697i)T \)
5 \( 1 + (-0.946 + 0.321i)T \)
7 \( 1 + (0.0261 + 0.999i)T \)
11 \( 1 + (-0.465 - 0.884i)T \)
13 \( 1 + (0.859 + 0.511i)T \)
17 \( 1 + (0.777 + 0.629i)T \)
19 \( 1 + (-0.801 - 0.598i)T \)
23 \( 1 + (-0.522 + 0.852i)T \)
29 \( 1 + (-0.785 + 0.619i)T \)
37 \( 1 + (-0.659 + 0.751i)T \)
41 \( 1 + (0.824 - 0.566i)T \)
43 \( 1 + (0.246 - 0.969i)T \)
47 \( 1 + (0.891 + 0.453i)T \)
53 \( 1 + (-0.985 + 0.169i)T \)
59 \( 1 + (-0.394 + 0.918i)T \)
61 \( 1 + (-0.555 + 0.831i)T \)
67 \( 1 + (0.997 + 0.0654i)T \)
71 \( 1 + (0.725 - 0.688i)T \)
73 \( 1 + (0.958 - 0.284i)T \)
79 \( 1 + (-0.629 + 0.777i)T \)
83 \( 1 + (0.918 - 0.394i)T \)
89 \( 1 + (0.522 + 0.852i)T \)
97 \( 1 + (0.587 - 0.809i)T \)
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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

−18.62648607388115937052680561710, −17.54938831944550497818066201792, −16.79401038814434404359538964110, −16.136207381066850129676440206, −15.755118687967879342778561410024, −14.9268980155118536881690292730, −14.402607364046154496414272434613, −13.676996023739331043290348829346, −12.80221830440773190465911772906, −12.40745247305398232093628747340, −11.14284069988597452107125693232, −10.80801739655306697619079492576, −10.00399195319427823397833098944, −9.42911811649012558242380356334, −8.42319451546087099529975321107, −7.84946917121492895817351120028, −7.5394705130765604591528700859, −6.46900082692222594588300845347, −5.32812221047906697854368696721, −4.59558268767689498558095662692, −3.96703154270807326294608552757, −3.51340271776718190380616556566, −2.53452526371334694110859740690, −1.51647446836662842613988340536, −0.33439128295978863100164279776, 1.03410554762077044188919955340, 1.97909738306274885843334163697, 2.79399786779023121731457636741, 3.512981053843054700319504093952, 4.02568120352585516659228835767, 5.34241520861995442951212481163, 6.07403064013747080720485658216, 6.715911084921122902096754319308, 7.67772378627274268611580073948, 8.118931041020372097829613236249, 8.83396243697480414094960304019, 9.23485879073917859051827336038, 10.56544343804584106901331767391, 11.132968762837777860049419550105, 11.9415851836476787147494923059, 12.4079117353826129381813905738, 13.176525625864686993013213013189, 13.91817221544151367624532822064, 14.52611651096734951366473727530, 15.42338194348297217629519735869, 15.56844847416797696698000242692, 16.48683400705706419004734667391, 17.425626440332248250805478733924, 18.359909779924268132181591639455, 18.74451053464115702345810695968

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