Properties

Degree 2
Conductor $ 2^{2} \cdot 7 \cdot 11 \cdot 13 $
Sign $0.978 - 0.206i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.08·3-s + 3.34i·5-s + i·7-s − 1.81·9-s i·11-s + (−3.52 + 0.743i)13-s − 3.64i·15-s + 4.31·17-s − 7.13i·19-s − 1.08i·21-s + 1.49·23-s − 6.22·25-s + 5.23·27-s + 2.80·29-s − 6.91i·31-s + ⋯
L(s)  = 1  − 0.627·3-s + 1.49i·5-s + 0.377i·7-s − 0.605·9-s − 0.301i·11-s + (−0.978 + 0.206i)13-s − 0.940i·15-s + 1.04·17-s − 1.63i·19-s − 0.237i·21-s + 0.310·23-s − 1.24·25-s + 1.00·27-s + 0.521·29-s − 1.24i·31-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.978 - 0.206i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
\( \varepsilon \)  =  $0.978 - 0.206i$
motivic weight  =  \(1\)
character  :  $\chi_{4004} (2157, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4004,\ (\ :1/2),\ 0.978 - 0.206i)$
$L(1)$  $\approx$  $1.120912941$
$L(\frac12)$  $\approx$  $1.120912941$
$L(\frac{3}{2})$   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 \( 1 + (3.52 - 0.743i)T \)
good3 \( 1 + 1.08T + 3T^{2} \)
5 \( 1 - 3.34iT - 5T^{2} \)
17 \( 1 - 4.31T + 17T^{2} \)
19 \( 1 + 7.13iT - 19T^{2} \)
23 \( 1 - 1.49T + 23T^{2} \)
29 \( 1 - 2.80T + 29T^{2} \)
31 \( 1 + 6.91iT - 31T^{2} \)
37 \( 1 - 3.56iT - 37T^{2} \)
41 \( 1 + 8.88iT - 41T^{2} \)
43 \( 1 + 6.06T + 43T^{2} \)
47 \( 1 + 4.61iT - 47T^{2} \)
53 \( 1 - 12.9T + 53T^{2} \)
59 \( 1 - 9.89iT - 59T^{2} \)
61 \( 1 - 3.97T + 61T^{2} \)
67 \( 1 + 12.7iT - 67T^{2} \)
71 \( 1 + 6.59iT - 71T^{2} \)
73 \( 1 - 12.0iT - 73T^{2} \)
79 \( 1 - 9.43T + 79T^{2} \)
83 \( 1 - 0.795iT - 83T^{2} \)
89 \( 1 + 1.02iT - 89T^{2} \)
97 \( 1 - 12.9iT - 97T^{2} \)
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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

−8.449539865812959098195087501963, −7.51649260688430815097898272625, −6.91817087906129695082110403585, −6.35736766574734731276387667297, −5.51853054192179413858080820619, −4.96287695718869850563464206618, −3.72079238313281581901300962292, −2.79945016934788766391007942140, −2.38302629352681937266527046694, −0.52263421017929306538365008878, 0.74995152367802405096194387057, 1.60424023969449084278281462679, 2.98376563086286443320626042515, 4.00093208732278177671928688897, 4.93984054407908137458447835867, 5.27667483252431675495893596035, 5.99868368593938969760479177704, 6.95411201699001304731388046872, 7.891752716285879028931759514020, 8.314984331964209047942735220171

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