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Properties

Label 2-4000-25.23-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.105 - 0.994i$
Analytic cond. $1.99626$
Root an. cond. $1.41289$
Motivic weight $0$
Arithmetic yes
Rational no
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

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.105 - 0.994i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.105 - 0.994i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.026883581\)
\(L(\frac12)\) \(\approx\) \(1.026883581\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
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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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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