Properties

Label 2-4004-13.12-c1-0-45
Degree $2$
Conductor $4004$
Sign $-0.361 + 0.932i$
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  − 1.07·3-s − 3.26i·5-s + i·7-s − 1.84·9-s i·11-s + (1.30 − 3.36i)13-s + 3.50i·15-s + 3.05·17-s + 3.15i·19-s − 1.07i·21-s + 5.49·23-s − 5.63·25-s + 5.20·27-s + 4.12·29-s − 6.96i·31-s + ⋯
L(s)  = 1  − 0.621·3-s − 1.45i·5-s + 0.377i·7-s − 0.614·9-s − 0.301i·11-s + (0.361 − 0.932i)13-s + 0.905i·15-s + 0.740·17-s + 0.724i·19-s − 0.234i·21-s + 1.14·23-s − 1.12·25-s + 1.00·27-s + 0.766·29-s − 1.25i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.361 + 0.932i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.361 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.320763303\)
\(L(\frac12)\) \(\approx\) \(1.320763303\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.30 + 3.36i)T \)
good3 \( 1 + 1.07T + 3T^{2} \)
5 \( 1 + 3.26iT - 5T^{2} \)
17 \( 1 - 3.05T + 17T^{2} \)
19 \( 1 - 3.15iT - 19T^{2} \)
23 \( 1 - 5.49T + 23T^{2} \)
29 \( 1 - 4.12T + 29T^{2} \)
31 \( 1 + 6.96iT - 31T^{2} \)
37 \( 1 - 10.1iT - 37T^{2} \)
41 \( 1 - 0.829iT - 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 + 9.75iT - 47T^{2} \)
53 \( 1 - 7.58T + 53T^{2} \)
59 \( 1 + 15.0iT - 59T^{2} \)
61 \( 1 + 14.7T + 61T^{2} \)
67 \( 1 - 9.94iT - 67T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 - 7.06iT - 73T^{2} \)
79 \( 1 - 11.7T + 79T^{2} \)
83 \( 1 + 0.295iT - 83T^{2} \)
89 \( 1 - 8.17iT - 89T^{2} \)
97 \( 1 + 9.48iT - 97T^{2} \)
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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

−8.265928503372400476162704077973, −7.78445485907508464975755910454, −6.49051672176223964527780720876, −5.78940122751823949095969023772, −5.30850304632614661560052681985, −4.72737509458732552191544390605, −3.61696793270652364710984626711, −2.70137019267261210021494513803, −1.26665997357694286251826888908, −0.52766596524892304750769902346, 1.06497103971239074793356454567, 2.51777834096143693378849808888, 3.10492560513614944198086989533, 4.09050637990791495997131913298, 4.98464463077447387485997512867, 5.89843580956891437007583129362, 6.49957544571526193354266123111, 7.14054689949637653471933057693, 7.61002762907543996522168723157, 8.842709618172817386223949210915

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