Properties

Label 2-4004-13.12-c1-0-2
Degree $2$
Conductor $4004$
Sign $-0.845 - 0.533i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.27·3-s − 2.17i·5-s + i·7-s + 7.72·9-s i·11-s + (3.04 + 1.92i)13-s + 7.13i·15-s − 4.89·17-s + 2.89i·19-s − 3.27i·21-s − 3.83·23-s + 0.253·25-s − 15.4·27-s + 3.96·29-s − 0.338i·31-s + ⋯
L(s)  = 1  − 1.89·3-s − 0.974i·5-s + 0.377i·7-s + 2.57·9-s − 0.301i·11-s + (0.845 + 0.533i)13-s + 1.84i·15-s − 1.18·17-s + 0.664i·19-s − 0.714i·21-s − 0.799·23-s + 0.0506·25-s − 2.97·27-s + 0.737·29-s − 0.0607i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.845 - 0.533i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.845 - 0.533i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.845 - 0.533i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.08286184607\)
\(L(\frac12)\) \(\approx\) \(0.08286184607\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.04 - 1.92i)T \)
good3 \( 1 + 3.27T + 3T^{2} \)
5 \( 1 + 2.17iT - 5T^{2} \)
17 \( 1 + 4.89T + 17T^{2} \)
19 \( 1 - 2.89iT - 19T^{2} \)
23 \( 1 + 3.83T + 23T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 + 0.338iT - 31T^{2} \)
37 \( 1 - 9.41iT - 37T^{2} \)
41 \( 1 + 1.25iT - 41T^{2} \)
43 \( 1 + 7.90T + 43T^{2} \)
47 \( 1 + 11.2iT - 47T^{2} \)
53 \( 1 + 1.19T + 53T^{2} \)
59 \( 1 + 9.32iT - 59T^{2} \)
61 \( 1 - 12.2T + 61T^{2} \)
67 \( 1 + 1.95iT - 67T^{2} \)
71 \( 1 - 5.38iT - 71T^{2} \)
73 \( 1 + 0.975iT - 73T^{2} \)
79 \( 1 - 5.42T + 79T^{2} \)
83 \( 1 - 4.61iT - 83T^{2} \)
89 \( 1 + 2.06iT - 89T^{2} \)
97 \( 1 + 0.930iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.594743884408823637889962398276, −8.217327958090179661062929059339, −6.79197851721027074042413389602, −6.53781659588358290379276910607, −5.72124515623545115327657446513, −5.09906824348899299490219135981, −4.49560496365340165975749066547, −3.72118080225812792018363855300, −1.93058276612862132516626153213, −1.08273211922506637129884620623, 0.03898014782673893639542785052, 1.19657908161039099101860782579, 2.46824424965573703839944385001, 3.76196057767867750227473356110, 4.46962525898277058670260452002, 5.24327492285090879806114995420, 6.11104122230072904573094955436, 6.55981015209233373890368615422, 7.07971238345531987497032609342, 7.85578275727355805011804307264

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