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Properties

Label 2-4004-13.12-c1-0-51
Degree $2$
Conductor $4004$
Sign $-0.134 + 0.990i$
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  + 0.726·3-s − 2.96i·5-s + i·7-s − 2.47·9-s + i·11-s + (−0.483 + 3.57i)13-s − 2.15i·15-s + 4.91·17-s − 6.32i·19-s + 0.726i·21-s + 0.763·23-s − 3.78·25-s − 3.97·27-s + 6.71·29-s + 5.63i·31-s + ⋯
L(s)  = 1  + 0.419·3-s − 1.32i·5-s + 0.377i·7-s − 0.824·9-s + 0.301i·11-s + (−0.134 + 0.990i)13-s − 0.555i·15-s + 1.19·17-s − 1.45i·19-s + 0.158i·21-s + 0.159·23-s − 0.756·25-s − 0.764·27-s + 1.24·29-s + 1.01i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.134 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.134 + 0.990i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.134 + 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.724418113\)
\(L(\frac12)\) \(\approx\) \(1.724418113\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.483 - 3.57i)T \)
good3 \( 1 - 0.726T + 3T^{2} \)
5 \( 1 + 2.96iT - 5T^{2} \)
17 \( 1 - 4.91T + 17T^{2} \)
19 \( 1 + 6.32iT - 19T^{2} \)
23 \( 1 - 0.763T + 23T^{2} \)
29 \( 1 - 6.71T + 29T^{2} \)
31 \( 1 - 5.63iT - 31T^{2} \)
37 \( 1 + 7.09iT - 37T^{2} \)
41 \( 1 + 7.75iT - 41T^{2} \)
43 \( 1 - 1.96T + 43T^{2} \)
47 \( 1 + 9.93iT - 47T^{2} \)
53 \( 1 + 6.03T + 53T^{2} \)
59 \( 1 + 4.32iT - 59T^{2} \)
61 \( 1 + 7.15T + 61T^{2} \)
67 \( 1 - 8.48iT - 67T^{2} \)
71 \( 1 + 8.48iT - 71T^{2} \)
73 \( 1 - 0.437iT - 73T^{2} \)
79 \( 1 - 2.91T + 79T^{2} \)
83 \( 1 + 16.2iT - 83T^{2} \)
89 \( 1 + 5.41iT - 89T^{2} \)
97 \( 1 - 9.24iT - 97T^{2} \)
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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

−8.526013415824818636976807110622, −7.63661096729837055916143404796, −6.86441388810338304930973717208, −5.89051962967875251722185611361, −5.10451852013264229052162732698, −4.66537842814760933251687483544, −3.58823085531756358509454563443, −2.65116667510227632070657597510, −1.70526392322307004078971214917, −0.50058026602099160479322276509, 1.16796763711185319593020414224, 2.64941692352490361194316983332, 3.09412482465112424161529273212, 3.71999683685769098038722469679, 4.93695823141018442633775407937, 6.03325854900386496218467326561, 6.20278276291704137065707301753, 7.38735108844615208747720750547, 7.966605127385341626093744751912, 8.297172886569840844035436577717

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