Properties

Label 2-4004-77.76-c1-0-86
Degree $2$
Conductor $4004$
Sign $-0.882 + 0.471i$
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  − 1.04i·3-s − 4.22i·5-s + (2.02 − 1.69i)7-s + 1.91·9-s + (3.24 + 0.678i)11-s − 13-s − 4.39·15-s + 3.87·17-s − 7.23·19-s + (−1.76 − 2.11i)21-s + 5.94·23-s − 12.8·25-s − 5.11i·27-s − 1.09i·29-s − 5.22i·31-s + ⋯
L(s)  = 1  − 0.600i·3-s − 1.88i·5-s + (0.766 − 0.641i)7-s + 0.638·9-s + (0.978 + 0.204i)11-s − 0.277·13-s − 1.13·15-s + 0.940·17-s − 1.66·19-s + (−0.385 − 0.460i)21-s + 1.23·23-s − 2.57·25-s − 0.984i·27-s − 0.203i·29-s − 0.937i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.471i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.882 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.882 + 0.471i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.882 + 0.471i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.323370647\)
\(L(\frac12)\) \(\approx\) \(2.323370647\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.02 + 1.69i)T \)
11 \( 1 + (-3.24 - 0.678i)T \)
13 \( 1 + T \)
good3 \( 1 + 1.04iT - 3T^{2} \)
5 \( 1 + 4.22iT - 5T^{2} \)
17 \( 1 - 3.87T + 17T^{2} \)
19 \( 1 + 7.23T + 19T^{2} \)
23 \( 1 - 5.94T + 23T^{2} \)
29 \( 1 + 1.09iT - 29T^{2} \)
31 \( 1 + 5.22iT - 31T^{2} \)
37 \( 1 - 3.09T + 37T^{2} \)
41 \( 1 + 2.53T + 41T^{2} \)
43 \( 1 - 6.88iT - 43T^{2} \)
47 \( 1 + 5.75iT - 47T^{2} \)
53 \( 1 - 8.66T + 53T^{2} \)
59 \( 1 - 6.74iT - 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 2.12T + 67T^{2} \)
71 \( 1 + 3.96T + 71T^{2} \)
73 \( 1 - 14.7T + 73T^{2} \)
79 \( 1 + 15.2iT - 79T^{2} \)
83 \( 1 - 3.04T + 83T^{2} \)
89 \( 1 - 7.88iT - 89T^{2} \)
97 \( 1 - 7.43iT - 97T^{2} \)
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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

−8.070344054210197443450249622331, −7.60199909699513268906813265971, −6.74636561870688830436265082561, −5.90708244893691522146512741170, −4.92517377116479009534511177251, −4.43304072030253067904851987116, −3.86085902689543797138248612598, −2.08740308209326649506418849823, −1.33554101346211660104833993726, −0.72659074955430908275541433547, 1.55539154343877626442665410255, 2.52860359283448758992021951343, 3.37014959502497774744761911121, 4.06766305201826503167190433881, 4.96213544481842922219845646313, 5.89174708465699434383644356785, 6.68450467760885963232686346662, 7.11269843229978889741194195096, 7.975587545752926909615066257613, 8.834948079795221477217466873475

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