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Properties

Label 2-4004-13.12-c1-0-50
Degree $2$
Conductor $4004$
Sign $0.649 - 0.760i$
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  + 2.94·3-s + 2.92i·5-s + i·7-s + 5.67·9-s + i·11-s + (2.34 − 2.74i)13-s + 8.62i·15-s + 4.73·17-s − 1.30i·19-s + 2.94i·21-s + 7.88·23-s − 3.58·25-s + 7.87·27-s + 6.34·29-s − 3.87i·31-s + ⋯
L(s)  = 1  + 1.70·3-s + 1.31i·5-s + 0.377i·7-s + 1.89·9-s + 0.301i·11-s + (0.649 − 0.760i)13-s + 2.22i·15-s + 1.14·17-s − 0.299i·19-s + 0.642i·21-s + 1.64·23-s − 0.716·25-s + 1.51·27-s + 1.17·29-s − 0.695i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.760i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.649 - 0.760i$
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: \((2,\ 4004,\ (\ :1/2),\ 0.649 - 0.760i)\)

Particular Values

\(L(1)\) \(\approx\) \(4.160390609\)
\(L(\frac12)\) \(\approx\) \(4.160390609\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.34 + 2.74i)T \)
good3 \( 1 - 2.94T + 3T^{2} \)
5 \( 1 - 2.92iT - 5T^{2} \)
17 \( 1 - 4.73T + 17T^{2} \)
19 \( 1 + 1.30iT - 19T^{2} \)
23 \( 1 - 7.88T + 23T^{2} \)
29 \( 1 - 6.34T + 29T^{2} \)
31 \( 1 + 3.87iT - 31T^{2} \)
37 \( 1 + 3.71iT - 37T^{2} \)
41 \( 1 + 7.91iT - 41T^{2} \)
43 \( 1 + 11.9T + 43T^{2} \)
47 \( 1 - 3.67iT - 47T^{2} \)
53 \( 1 + 9.14T + 53T^{2} \)
59 \( 1 - 14.1iT - 59T^{2} \)
61 \( 1 + 13.6T + 61T^{2} \)
67 \( 1 - 13.5iT - 67T^{2} \)
71 \( 1 + 8.37iT - 71T^{2} \)
73 \( 1 + 3.65iT - 73T^{2} \)
79 \( 1 + 8.06T + 79T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + 13.2iT - 89T^{2} \)
97 \( 1 - 9.09iT - 97T^{2} \)
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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

−8.582623208462415822517991203922, −7.77108759116770162041089386077, −7.31850246556246219534205135114, −6.59470523901460923521613376991, −5.66454265826383786431113124298, −4.57307861650835613412569511453, −3.46145328632918251427453124448, −3.06751288313812699696281647515, −2.51871670575625754091072191943, −1.34693122574795978561922816623, 1.14656354473231015302755231794, 1.64269353142940256525847350030, 3.13723401107902693403148472523, 3.42297668467640162668086528272, 4.61651283156277685589615038122, 4.94300584534262709395975189741, 6.29554008029134149589490140459, 7.07029233707998334524912239109, 8.042118175524870719927002072647, 8.333368156974550037063973905177

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