Properties

Degree 2
Conductor $ 2^{5} \cdot 5^{3} $
Sign $0.105 + 0.994i$
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 + (0.142 − 0.896i)13-s + (−0.278 + 0.142i)17-s + (1.11 − 0.363i)29-s + (−1.95 − 0.309i)37-s + (1.53 − 1.11i)41-s i·49-s + (−1.58 − 0.809i)53-s + (−1.53 − 1.11i)61-s + (1.76 − 0.278i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.896 + 1.76i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)9-s + (0.142 − 0.896i)13-s + (−0.278 + 0.142i)17-s + (1.11 − 0.363i)29-s + (−1.95 − 0.309i)37-s + (1.53 − 1.11i)41-s i·49-s + (−1.58 − 0.809i)53-s + (−1.53 − 1.11i)61-s + (1.76 − 0.278i)73-s + (−0.309 + 0.951i)81-s + (0.690 − 0.951i)89-s + (−0.896 + 1.76i)97-s + 1.61·101-s + (0.363 + 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.105 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4000\)    =    \(2^{5} \cdot 5^{3}\)
\( \varepsilon \)  =  $0.105 + 0.994i$
motivic weight  =  \(0\)
character  :  $\chi_{4000} (3457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4000,\ (\ :0),\ 0.105 + 0.994i)$
$L(\frac{1}{2})$  $\approx$  $1.026883581$
$L(\frac12)$  $\approx$  $1.026883581$
$L(1)$   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 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
17 \( 1 + (0.278 - 0.142i)T + (0.587 - 0.809i)T^{2} \)
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 \( 1 + (1.95 + 0.309i)T + (0.951 + 0.309i)T^{2} \)
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 \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \)
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 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
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 \( 1 + (0.896 - 1.76i)T + (-0.587 - 0.809i)T^{2} \)
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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

−8.453743268200076429784660292229, −7.84848210786927213243968838719, −6.89533397988922058432011917778, −6.25662276746199384674519340248, −5.53936783410052730666600455816, −4.75299605544169250106007051307, −3.67651181652074372227123776237, −3.09868494906398745957963937224, −2.00469962958422626197963829526, −0.57895258705990064752217201139, 1.43870518205962038865426797308, 2.46867510874005352708696823966, 3.27606640128429834635803549029, 4.44697136689052723308720081412, 4.91242802257040244233212988725, 5.92422512567437651674502343991, 6.54302245244872140342536079358, 7.39956522403477148091255275728, 8.074462164533178584142168928965, 8.823471277066082851244550107488

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