Properties

Label 2-4004-13.12-c1-0-60
Degree $2$
Conductor $4004$
Sign $0.262 + 0.965i$
Analytic cond. $31.9721$
Root an. cond. $5.65438$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.18·3-s − 0.656i·5-s + i·7-s + 1.76·9-s i·11-s + (−0.945 − 3.47i)13-s − 1.43i·15-s + 2.46·17-s − 5.98i·19-s + 2.18i·21-s − 1.56·23-s + 4.56·25-s − 2.70·27-s + 1.39·29-s − 7.01i·31-s + ⋯
L(s)  = 1  + 1.25·3-s − 0.293i·5-s + 0.377i·7-s + 0.586·9-s − 0.301i·11-s + (−0.262 − 0.965i)13-s − 0.369i·15-s + 0.596·17-s − 1.37i·19-s + 0.476i·21-s − 0.326·23-s + 0.913·25-s − 0.520·27-s + 0.259·29-s − 1.25i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.262 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.262 + 0.965i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (2157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 0.262 + 0.965i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.662375369\)
\(L(\frac12)\) \(\approx\) \(2.662375369\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - iT \)
11 \( 1 + iT \)
13 \( 1 + (0.945 + 3.47i)T \)
good3 \( 1 - 2.18T + 3T^{2} \)
5 \( 1 + 0.656iT - 5T^{2} \)
17 \( 1 - 2.46T + 17T^{2} \)
19 \( 1 + 5.98iT - 19T^{2} \)
23 \( 1 + 1.56T + 23T^{2} \)
29 \( 1 - 1.39T + 29T^{2} \)
31 \( 1 + 7.01iT - 31T^{2} \)
37 \( 1 - 6.51iT - 37T^{2} \)
41 \( 1 + 1.43iT - 41T^{2} \)
43 \( 1 + 2.49T + 43T^{2} \)
47 \( 1 + 2.52iT - 47T^{2} \)
53 \( 1 + 7.93T + 53T^{2} \)
59 \( 1 + 6.65iT - 59T^{2} \)
61 \( 1 - 9.05T + 61T^{2} \)
67 \( 1 + 1.47iT - 67T^{2} \)
71 \( 1 - 4.89iT - 71T^{2} \)
73 \( 1 + 0.638iT - 73T^{2} \)
79 \( 1 + 1.86T + 79T^{2} \)
83 \( 1 + 9.14iT - 83T^{2} \)
89 \( 1 + 8.47iT - 89T^{2} \)
97 \( 1 + 9.88iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.384248383148156660921901846011, −7.79960158858807226024789220725, −7.03090004525446559208986099682, −6.05636204177848766287053471708, −5.23295189110049131660212900474, −4.47988597599685812470734495099, −3.32026229041227983426378541684, −2.90192579958384115715345824960, −2.00426072064284423414441806018, −0.62760279284974971867946858906, 1.40301244150890045035112081413, 2.25845807826510091440332987777, 3.19530755358613899793882509580, 3.80608892128824917032743736455, 4.63251509800755293368965506465, 5.65295759696717488669359934460, 6.62947227415885120998979556445, 7.26244143643524069873522226427, 7.946949365505132129973073210412, 8.535787460579331146781263174224

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