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Properties

Label 2-4004-13.12-c1-0-28
Degree $2$
Conductor $4004$
Sign $-0.408 - 0.912i$
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  + 2.43·3-s + 3.70i·5-s + i·7-s + 2.95·9-s − i·11-s + (1.47 + 3.29i)13-s + 9.03i·15-s + 0.503·17-s − 0.759i·19-s + 2.43i·21-s + 7.83·23-s − 8.70·25-s − 0.114·27-s − 8.07·29-s + 3.66i·31-s + ⋯
L(s)  = 1  + 1.40·3-s + 1.65i·5-s + 0.377i·7-s + 0.984·9-s − 0.301i·11-s + (0.408 + 0.912i)13-s + 2.33i·15-s + 0.122·17-s − 0.174i·19-s + 0.532i·21-s + 1.63·23-s − 1.74·25-s − 0.0219·27-s − 1.50·29-s + 0.657i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.408 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.408 - 0.912i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.408 - 0.912i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.027134119\)
\(L(\frac12)\) \(\approx\) \(3.027134119\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 3.29i)T \)
good3 \( 1 - 2.43T + 3T^{2} \)
5 \( 1 - 3.70iT - 5T^{2} \)
17 \( 1 - 0.503T + 17T^{2} \)
19 \( 1 + 0.759iT - 19T^{2} \)
23 \( 1 - 7.83T + 23T^{2} \)
29 \( 1 + 8.07T + 29T^{2} \)
31 \( 1 - 3.66iT - 31T^{2} \)
37 \( 1 - 5.15iT - 37T^{2} \)
41 \( 1 - 11.3iT - 41T^{2} \)
43 \( 1 + 2.20T + 43T^{2} \)
47 \( 1 + 3.86iT - 47T^{2} \)
53 \( 1 - 7.41T + 53T^{2} \)
59 \( 1 + 3.16iT - 59T^{2} \)
61 \( 1 - 4.30T + 61T^{2} \)
67 \( 1 + 14.7iT - 67T^{2} \)
71 \( 1 - 5.64iT - 71T^{2} \)
73 \( 1 + 14.4iT - 73T^{2} \)
79 \( 1 + 3.53T + 79T^{2} \)
83 \( 1 - 8.51iT - 83T^{2} \)
89 \( 1 + 13.3iT - 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−8.684411583038151611518208848503, −7.985806106402633874619664094418, −7.18284417856303150768453982097, −6.74537967523022621969848355629, −5.93832801924192494129450690051, −4.78743047966432161416424834712, −3.59764196712166175110742881056, −3.23839821561361199027752574243, −2.51873383641577025127455993445, −1.67307536543779307174350580812, 0.69349062186293089139878562972, 1.64819333201782656764937585932, 2.64039990867189308977323271940, 3.71102430265703759334691635400, 4.15730428305469291123610317342, 5.25322243362815150468226720687, 5.68947032349815069384855465658, 7.19110994761504050644048206586, 7.60795194435808551104648420485, 8.443507195216155402615009929364

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