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Properties

Label 1-3968-3968.1091-r0-0-0
Degree $1$
Conductor $3968$
Sign $-0.947 - 0.320i$
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  + (0.751 + 0.659i)3-s + (−0.0654 + 0.997i)5-s + (0.130 − 0.991i)7-s + (0.130 + 0.991i)9-s + (0.659 + 0.751i)11-s + (0.896 + 0.442i)13-s + (−0.707 + 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.997 + 0.0654i)19-s + (0.751 − 0.659i)21-s + (−0.382 + 0.923i)23-s + (−0.991 − 0.130i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s + i·33-s + ⋯
L(s)  = 1  + (0.751 + 0.659i)3-s + (−0.0654 + 0.997i)5-s + (0.130 − 0.991i)7-s + (0.130 + 0.991i)9-s + (0.659 + 0.751i)11-s + (0.896 + 0.442i)13-s + (−0.707 + 0.707i)15-s + (−0.965 + 0.258i)17-s + (−0.997 + 0.0654i)19-s + (0.751 − 0.659i)21-s + (−0.382 + 0.923i)23-s + (−0.991 − 0.130i)25-s + (−0.555 + 0.831i)27-s + (−0.980 − 0.195i)29-s + i·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.947 - 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $-0.947 - 0.320i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1091, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ -0.947 - 0.320i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1735332732 + 1.052676854i\)
\(L(\frac12)\) \(\approx\) \(-0.1735332732 + 1.052676854i\)
\(L(1)\) \(\approx\) \(1.007255993 + 0.5493990305i\)
\(L(1)\) \(\approx\) \(1.007255993 + 0.5493990305i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (0.751 + 0.659i)T \)
5 \( 1 + (-0.0654 + 0.997i)T \)
7 \( 1 + (0.130 - 0.991i)T \)
11 \( 1 + (0.659 + 0.751i)T \)
13 \( 1 + (0.896 + 0.442i)T \)
17 \( 1 + (-0.965 + 0.258i)T \)
19 \( 1 + (-0.997 + 0.0654i)T \)
23 \( 1 + (-0.382 + 0.923i)T \)
29 \( 1 + (-0.980 - 0.195i)T \)
37 \( 1 + (-0.442 - 0.896i)T \)
41 \( 1 + (-0.991 + 0.130i)T \)
43 \( 1 + (-0.946 - 0.321i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.659 + 0.751i)T \)
59 \( 1 + (0.896 - 0.442i)T \)
61 \( 1 + (0.195 - 0.980i)T \)
67 \( 1 + (-0.946 + 0.321i)T \)
71 \( 1 + (-0.793 - 0.608i)T \)
73 \( 1 + (0.130 + 0.991i)T \)
79 \( 1 + (0.258 + 0.965i)T \)
83 \( 1 + (0.442 - 0.896i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
97 \( 1 - iT \)
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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

−18.11843262334963587100583811501, −17.67363528872499466976310329664, −16.652000075786661738875621591411, −16.13595549081243287280470086484, −15.18589252217936189161333535852, −14.888882275775702756226175431240, −13.864630139634337728621624077868, −13.16609265337007701299232241681, −12.91519106461343245935518623975, −11.88228562535609140685066608353, −11.61559122785382757255543588958, −10.52162370521168232385517847129, −9.438206278480133274790675425722, −8.69888904167224217964133122174, −8.61268905480344655844547207979, −7.96188315170717164683183178656, −6.7089506354343522548793161021, −6.23612790686573689737647477262, −5.452102570807763215153796404893, −4.47933502392779855295815387603, −3.7113682256462498973529867443, −2.88082689407910689798412454178, −1.94490690108364613689880816015, −1.38728710819298395663027879976, −0.23587399693919145150812012727, 1.72803077948924087878549950298, 2.03135690362142660246513876967, 3.28982512406852324680497017744, 3.99656580726601442578503406495, 4.13350144618344852249459843035, 5.31818746490553572108928721118, 6.48859516535406496110258635188, 6.92368028161924385208244073814, 7.69099038511739096756357836688, 8.46230871427412944736292167638, 9.22155689918461013537358510873, 9.961137950835437616778693418960, 10.54708512764141121232293418005, 11.12400800220280973536841450893, 11.73821346289938197141590699824, 13.073396644013397106618055889736, 13.55077077936350783871125904674, 14.15658880653194436338048912491, 14.89853032079567050755583033169, 15.22637392313290663432135475333, 16.0499231216451185571089964182, 16.84261958450479493027653704409, 17.47138025193078961251813733849, 18.201151579425842136872902397183, 19.07880537017175851635713250639

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