Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $-0.361 - 0.932i$
Motivic weight 1
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

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $-0.361 - 0.932i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ -0.361 - 0.932i)$
$L(1)$  $\approx$  $1.320763303$
$L(\frac12)$  $\approx$  $1.320763303$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;7,\;11,\;13\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
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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\[\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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