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Properties

Label 2-4000-25.3-c0-0-2
Degree $2$
Conductor $4000$
Sign $0.603 + 0.797i$
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  + (0.951 − 0.309i)9-s + (0.896 − 1.76i)13-s + (0.142 − 0.896i)17-s + (−1.11 + 1.53i)29-s + (−1.58 − 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (−0.0489 − 0.309i)53-s + (0.363 − 1.11i)61-s + (0.278 − 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (1.76 − 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯
L(s)  = 1  + (0.951 − 0.309i)9-s + (0.896 − 1.76i)13-s + (0.142 − 0.896i)17-s + (−1.11 + 1.53i)29-s + (−1.58 − 0.809i)37-s + (−0.363 − 1.11i)41-s + i·49-s + (−0.0489 − 0.309i)53-s + (0.363 − 1.11i)61-s + (0.278 − 0.142i)73-s + (0.809 − 0.587i)81-s + (1.80 + 0.587i)89-s + (1.76 − 0.278i)97-s − 0.618·101-s + (1.53 − 0.5i)109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.603 + 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.603 + 0.797i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (993, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.603 + 0.797i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.393337988\)
\(L(\frac12)\) \(\approx\) \(1.393337988\)
\(L(1)\) 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 \( 1 + (-0.951 + 0.309i)T^{2} \)
7 \( 1 - iT^{2} \)
11 \( 1 + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.896 + 1.76i)T + (-0.587 - 0.809i)T^{2} \)
17 \( 1 + (-0.142 + 0.896i)T + (-0.951 - 0.309i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (0.587 - 0.809i)T^{2} \)
29 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
31 \( 1 + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (1.58 + 0.809i)T + (0.587 + 0.809i)T^{2} \)
41 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (0.951 - 0.309i)T^{2} \)
53 \( 1 + (0.0489 + 0.309i)T + (-0.951 + 0.309i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
67 \( 1 + (-0.951 - 0.309i)T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.278 + 0.142i)T + (0.587 - 0.809i)T^{2} \)
79 \( 1 + (-0.309 - 0.951i)T^{2} \)
83 \( 1 + (0.951 + 0.309i)T^{2} \)
89 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-1.76 + 0.278i)T + (0.951 - 0.309i)T^{2} \)
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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

−8.599773307452145536252978837235, −7.61356939748501633282296986458, −7.22967374809209278433798129336, −6.32234070202906041104224057211, −5.44445313270051741933646964242, −4.93413493819491911179388056416, −3.64287339607775174670315998273, −3.32809014689970200764436480556, −1.96819925723893023740312552428, −0.842515910366969655842917993558, 1.48825040359220765606321554022, 2.07116336530858488929950858830, 3.55697033631481768419599102173, 4.12418432105970475336511024371, 4.84451652299861274245209553398, 5.92281261241630302735032277258, 6.56457647911929271893794644833, 7.19510400631460177604893541842, 8.047808839199457332303873745098, 8.714026072837800928149217760289

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