Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.349 + 0.936i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.587 − 0.809i)9-s + (−1.76 + 0.278i)13-s + (0.896 − 1.76i)17-s + (1.11 + 0.363i)29-s + (−0.0489 − 0.309i)37-s + (−1.53 − 1.11i)41-s i·49-s + (−0.412 − 0.809i)53-s + (1.53 − 1.11i)61-s + (−0.142 + 0.896i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯
L(s)  = 1  + (0.587 − 0.809i)9-s + (−1.76 + 0.278i)13-s + (0.896 − 1.76i)17-s + (1.11 + 0.363i)29-s + (−0.0489 − 0.309i)37-s + (−1.53 − 1.11i)41-s i·49-s + (−0.412 − 0.809i)53-s + (1.53 − 1.11i)61-s + (−0.142 + 0.896i)73-s + (−0.309 − 0.951i)81-s + (0.690 + 0.951i)89-s + (0.278 − 0.142i)97-s + 1.61·101-s + (−0.363 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.349 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.349 + 0.936i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (2657, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.349 + 0.936i)$
$L(\frac{1}{2})$  $\approx$  $1.155501637$
$L(\frac12)$  $\approx$  $1.155501637$
$L(1)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;5\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
5 \( 1 \)
good3 \( 1 + (-0.587 + 0.809i)T^{2} \)
7 \( 1 + iT^{2} \)
11 \( 1 + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (1.76 - 0.278i)T + (0.951 - 0.309i)T^{2} \)
17 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.951 - 0.309i)T^{2} \)
29 \( 1 + (-1.11 - 0.363i)T + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \)
41 \( 1 + (1.53 + 1.11i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + (0.587 - 0.809i)T^{2} \)
53 \( 1 + (0.412 + 0.809i)T + (-0.587 + 0.809i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-1.53 + 1.11i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (-0.587 - 0.809i)T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (0.142 - 0.896i)T + (-0.951 - 0.309i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.587 + 0.809i)T^{2} \)
89 \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \)
97 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
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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

−8.540618750952022167753430638381, −7.54779036353424888098306991706, −7.05468925532623049434020511854, −6.52615789206189107973587363491, −5.14174828721281966561032318406, −5.01509339661478388007893203552, −3.83364520196653666878542299588, −3.00156338490065672218981955566, −2.07700804261814804163447586923, −0.66706664084150803662014498185, 1.40918208346695072936083268714, 2.36816929672583738871697376756, 3.29099709857617645181734339917, 4.39794545880960487511978560322, 4.91637179239567123045069275797, 5.77885180374325689918825911387, 6.60615365631797653007112430295, 7.46655943476229902502618431901, 7.945961478198880127290574118842, 8.594495559515315754026730682364

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