Properties

Label 2-4004-77.76-c1-0-93
Degree $2$
Conductor $4004$
Sign $0.236 - 0.971i$
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  − 2.39i·3-s − 2.83i·5-s + (−2.55 − 0.678i)7-s − 2.74·9-s + (−0.0667 − 3.31i)11-s − 13-s − 6.79·15-s − 4.04·17-s − 4.55·19-s + (−1.62 + 6.13i)21-s + 5.77·23-s − 3.02·25-s − 0.607i·27-s − 4.69i·29-s + 1.01i·31-s + ⋯
L(s)  = 1  − 1.38i·3-s − 1.26i·5-s + (−0.966 − 0.256i)7-s − 0.915·9-s + (−0.0201 − 0.999i)11-s − 0.277·13-s − 1.75·15-s − 0.980·17-s − 1.04·19-s + (−0.354 + 1.33i)21-s + 1.20·23-s − 0.605·25-s − 0.116i·27-s − 0.872i·29-s + 0.182i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.236 - 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.236 - 0.971i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (3849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 0.236 - 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6879239680\)
\(L(\frac12)\) \(\approx\) \(0.6879239680\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.55 + 0.678i)T \)
11 \( 1 + (0.0667 + 3.31i)T \)
13 \( 1 + T \)
good3 \( 1 + 2.39iT - 3T^{2} \)
5 \( 1 + 2.83iT - 5T^{2} \)
17 \( 1 + 4.04T + 17T^{2} \)
19 \( 1 + 4.55T + 19T^{2} \)
23 \( 1 - 5.77T + 23T^{2} \)
29 \( 1 + 4.69iT - 29T^{2} \)
31 \( 1 - 1.01iT - 31T^{2} \)
37 \( 1 + 1.21T + 37T^{2} \)
41 \( 1 + 0.889T + 41T^{2} \)
43 \( 1 - 1.03iT - 43T^{2} \)
47 \( 1 + 4.60iT - 47T^{2} \)
53 \( 1 + 7.85T + 53T^{2} \)
59 \( 1 - 2.22iT - 59T^{2} \)
61 \( 1 + 6.50T + 61T^{2} \)
67 \( 1 - 8.04T + 67T^{2} \)
71 \( 1 - 10.7T + 71T^{2} \)
73 \( 1 - 1.88T + 73T^{2} \)
79 \( 1 - 11.9iT - 79T^{2} \)
83 \( 1 + 2.58T + 83T^{2} \)
89 \( 1 + 14.4iT - 89T^{2} \)
97 \( 1 - 2.52iT - 97T^{2} \)
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

−7.949384758509914517233726248463, −7.00326100019287204298093137467, −6.51441397391532828237228476981, −5.85943534015275386141848533974, −4.92052724855135816599221771466, −4.06086847652372157551075795516, −2.96046000405810148889032602310, −2.02328411457768460185856368533, −0.937316127908806073410086212219, −0.22939482187071551363664238289, 2.14996264789878100423407547424, 2.96611047806333596610879789668, 3.59941990508863911188756709265, 4.48253593911312702591353955443, 5.08241864976835285418122680704, 6.21602844300918696752942014714, 6.78331869102781369059662619256, 7.31475091485489885149443288287, 8.552776551319143200007344239482, 9.298522515343707187570232383368

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