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Properties

Label 1-4008-4008.29-r1-0-0
Degree $1$
Conductor $4008$
Sign $0.983 - 0.181i$
Analytic cond. $430.719$
Root an. cond. $430.719$
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  + (0.822 − 0.569i)5-s + (−0.700 − 0.713i)7-s + (−0.316 + 0.948i)11-s + (−0.898 − 0.438i)13-s + (−0.776 + 0.629i)17-s + (0.843 − 0.537i)19-s + (0.965 − 0.261i)23-s + (0.351 − 0.936i)25-s + (−0.965 − 0.261i)29-s + (−0.455 + 0.890i)31-s + (−0.982 − 0.188i)35-s + (−0.726 − 0.686i)37-s + (0.584 + 0.811i)41-s + (0.752 + 0.658i)43-s + (0.169 − 0.985i)47-s + ⋯
L(s)  = 1  + (0.822 − 0.569i)5-s + (−0.700 − 0.713i)7-s + (−0.316 + 0.948i)11-s + (−0.898 − 0.438i)13-s + (−0.776 + 0.629i)17-s + (0.843 − 0.537i)19-s + (0.965 − 0.261i)23-s + (0.351 − 0.936i)25-s + (−0.965 − 0.261i)29-s + (−0.455 + 0.890i)31-s + (−0.982 − 0.188i)35-s + (−0.726 − 0.686i)37-s + (0.584 + 0.811i)41-s + (0.752 + 0.658i)43-s + (0.169 − 0.985i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4008 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.983 - 0.181i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4008\)    =    \(2^{3} \cdot 3 \cdot 167\)
Sign: $0.983 - 0.181i$
Analytic conductor: \(430.719\)
Root analytic conductor: \(430.719\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4008} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4008,\ (1:\ ),\ 0.983 - 0.181i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.704607209 - 0.1558417931i\)
\(L(\frac12)\) \(\approx\) \(1.704607209 - 0.1558417931i\)
\(L(1)\) \(\approx\) \(0.9998914156 - 0.1346758535i\)
\(L(1)\) \(\approx\) \(0.9998914156 - 0.1346758535i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
167 \( 1 \)
good5 \( 1 + (0.822 - 0.569i)T \)
7 \( 1 + (-0.700 - 0.713i)T \)
11 \( 1 + (-0.316 + 0.948i)T \)
13 \( 1 + (-0.898 - 0.438i)T \)
17 \( 1 + (-0.776 + 0.629i)T \)
19 \( 1 + (0.843 - 0.537i)T \)
23 \( 1 + (0.965 - 0.261i)T \)
29 \( 1 + (-0.965 - 0.261i)T \)
31 \( 1 + (-0.455 + 0.890i)T \)
37 \( 1 + (-0.726 - 0.686i)T \)
41 \( 1 + (0.584 + 0.811i)T \)
43 \( 1 + (0.752 + 0.658i)T \)
47 \( 1 + (0.169 - 0.985i)T \)
53 \( 1 + (-0.999 + 0.0378i)T \)
59 \( 1 + (0.776 + 0.629i)T \)
61 \( 1 + (0.644 + 0.764i)T \)
67 \( 1 + (-0.822 - 0.569i)T \)
71 \( 1 + (0.521 + 0.853i)T \)
73 \( 1 + (-0.881 + 0.472i)T \)
79 \( 1 + (-0.0944 - 0.995i)T \)
83 \( 1 + (0.614 - 0.788i)T \)
89 \( 1 + (-0.553 - 0.832i)T \)
97 \( 1 + (-0.455 - 0.890i)T \)
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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

−18.46078507978911677882357416133, −17.67552491260715416745909443084, −16.99371733224565450140831176897, −16.28846504074736911675395191278, −15.634231427242282894110129577614, −14.9086695316922811168012451594, −14.16801491641356290499547515027, −13.59813258509888065169117074434, −12.94253931942840279705801116909, −12.213316755077049241866375046946, −11.30759051174743247422068450707, −10.8620026817071405261240399134, −9.82710325888160483699594989288, −9.3667445677714373002203324286, −8.8795889526344300519029327497, −7.71341265845486624401752150904, −7.008740242872665836813063697700, −6.3618680479266660180235979748, −5.50137535009287222905204799063, −5.2050095825386894164051838376, −3.85709029431255352830786252051, −3.00490151646633890771140197574, −2.52348595096709921781863405037, −1.67936596615966447074019524143, −0.411436839172028859641876982733, 0.50780535326992448895298288699, 1.43993655914529152381749428546, 2.32794440331304078062873698607, 3.04732269236400059857904130212, 4.12827082549638618066955353296, 4.837780111731919004271942880090, 5.44615753164918842090051804056, 6.347019404874343875092522432202, 7.16685814475569759485481359843, 7.551112985268553633472644132803, 8.801077622361508885927249744935, 9.29201245809829752927119981416, 10.035153801449535077499072134851, 10.45252530280841540974824205412, 11.356862185803409951830615490545, 12.437931536316940572878535030425, 12.93596938875821264643841556311, 13.238672282904728749537879665348, 14.19892374827624502349387062104, 14.83990915356818124079963801108, 15.64534997381049152235209503111, 16.34072064334308869403784609457, 17.01906935591738075515219246605, 17.64188527680766650446566272164, 17.943626955927622315413727887604

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