↑

Properties

Label 1-3968-3968.1331-r0-0-0
Degree $1$
Conductor $3968$
Sign $-0.227 - 0.973i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.271 − 0.962i)3-s + (0.980 − 0.195i)5-s + (−0.852 − 0.522i)7-s + (−0.852 + 0.522i)9-s + (−0.0392 − 0.999i)11-s + (0.488 − 0.872i)13-s + (−0.453 − 0.891i)15-s + (0.156 + 0.987i)17-s + (0.993 − 0.117i)19-s + (−0.271 + 0.962i)21-s + (0.522 + 0.852i)23-s + (0.923 − 0.382i)25-s + (0.734 + 0.678i)27-s + (0.619 − 0.785i)29-s + (−0.951 + 0.309i)33-s + ⋯
L(s)  = 1  + (−0.271 − 0.962i)3-s + (0.980 − 0.195i)5-s + (−0.852 − 0.522i)7-s + (−0.852 + 0.522i)9-s + (−0.0392 − 0.999i)11-s + (0.488 − 0.872i)13-s + (−0.453 − 0.891i)15-s + (0.156 + 0.987i)17-s + (0.993 − 0.117i)19-s + (−0.271 + 0.962i)21-s + (0.522 + 0.852i)23-s + (0.923 − 0.382i)25-s + (0.734 + 0.678i)27-s + (0.619 − 0.785i)29-s + (−0.951 + 0.309i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3968 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.227 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3968\)    =    \(2^{7} \cdot 31\)
Sign: $-0.227 - 0.973i$
Analytic conductor: \(18.4273\)
Root analytic conductor: \(18.4273\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3968} (1331, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3968,\ (0:\ ),\ -0.227 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.193283550 - 1.503540516i\)
\(L(\frac12)\) \(\approx\) \(1.193283550 - 1.503540516i\)
\(L(1)\) \(\approx\) \(1.027591502 - 0.5446326943i\)
\(L(1)\) \(\approx\) \(1.027591502 - 0.5446326943i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.271 - 0.962i)T \)
5 \( 1 + (0.980 - 0.195i)T \)
7 \( 1 + (-0.852 - 0.522i)T \)
11 \( 1 + (-0.0392 - 0.999i)T \)
13 \( 1 + (0.488 - 0.872i)T \)
17 \( 1 + (0.156 + 0.987i)T \)
19 \( 1 + (0.993 - 0.117i)T \)
23 \( 1 + (0.522 + 0.852i)T \)
29 \( 1 + (0.619 - 0.785i)T \)
37 \( 1 + (-0.195 - 0.980i)T \)
41 \( 1 + (-0.0784 + 0.996i)T \)
43 \( 1 + (-0.271 + 0.962i)T \)
47 \( 1 + (0.891 - 0.453i)T \)
53 \( 1 + (0.938 - 0.346i)T \)
59 \( 1 + (0.117 - 0.993i)T \)
61 \( 1 + (0.831 - 0.555i)T \)
67 \( 1 + (0.831 - 0.555i)T \)
71 \( 1 + (-0.233 + 0.972i)T \)
73 \( 1 + (0.233 + 0.972i)T \)
79 \( 1 + (0.987 - 0.156i)T \)
83 \( 1 + (0.993 - 0.117i)T \)
89 \( 1 + (-0.522 + 0.852i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.41656550685869916743471367353, −18.13098172137295364926126638876, −17.20565701443745574222664613396, −16.57606651355058020141105602506, −16.05040720179826584773800578381, −15.37278576134638591821148025140, −14.67104590314692037453670189277, −13.92656861777037657483845256103, −13.41982752872739969552843150107, −12.23556863537288297267161523398, −12.01122913165250516099499132575, −10.933452228811869456082866475930, −10.21177262253754197626728923570, −9.77006467446387777203133840131, −9.06520371636893156068524528597, −8.739212711113678373501989229697, −7.14293958205145008236688184384, −6.72324553435371307760004353353, −5.867962651450771591663420317012, −5.22214080990970680804455667457, −4.605834814170940694508346031106, −3.58338951036447928237338931330, −2.837884707760055494296478580939, −2.17544061686495214298966984027, −0.93668226287778787934741216087, 0.764838793319559214982422096136, 1.14668952601252404969904543339, 2.2889168030889191145902042950, 3.09430999947127770606293594476, 3.73088284214907054488996552339, 5.202410404712237654149055454198, 5.68579191083135956633227060622, 6.291925626827072643423746226563, 6.87656601346779315766617881319, 7.8608989330025182282530808519, 8.39398390973759826456516067102, 9.31449131628607326489945642298, 10.01730882198425058544093418722, 10.771101442754094022481039669619, 11.35577145398973529883512079081, 12.37330077163591258080458534898, 12.984658100078711631196999873157, 13.47263702095090738275806225029, 13.835010515326439462310555665234, 14.7119039269500247331599922571, 15.82697762562531145148305222049, 16.37561958351318721739695895887, 17.16000523086818715657132586199, 17.5360286910032116472733998925, 18.2759779481127854347974734018

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