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Properties

Label 2-4000-5.3-c0-0-7
Degree $2$
Conductor $4000$
Sign $0.707 + 0.707i$
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  + (0.221 + 0.221i)3-s + (1.26 − 1.26i)7-s − 0.902i·9-s + 0.557·21-s + (0.642 + 0.642i)23-s + (0.420 − 0.420i)27-s + 1.61i·29-s − 1.17·41-s + (−1.39 − 1.39i)43-s + (1.39 − 1.39i)47-s − 2.17i·49-s − 1.90·61-s + (−1.13 − 1.13i)63-s + (1 − i)67-s + 0.284i·69-s + ⋯
L(s)  = 1  + (0.221 + 0.221i)3-s + (1.26 − 1.26i)7-s − 0.902i·9-s + 0.557·21-s + (0.642 + 0.642i)23-s + (0.420 − 0.420i)27-s + 1.61i·29-s − 1.17·41-s + (−1.39 − 1.39i)43-s + (1.39 − 1.39i)47-s − 2.17i·49-s − 1.90·61-s + (−1.13 − 1.13i)63-s + (1 − i)67-s + 0.284i·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.707 + 0.707i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.707 + 0.707i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.624462621\)
\(L(\frac12)\) \(\approx\) \(1.624462621\)
\(L(1)\) 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 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
7 \( 1 + (-1.26 + 1.26i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
29 \( 1 - 1.61iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 + 1.17T + T^{2} \)
43 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
47 \( 1 + (-1.39 + 1.39i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.90T + T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
89 \( 1 - 0.618iT - T^{2} \)
97 \( 1 - iT^{2} \)
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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

−8.633808838135999941808845730093, −7.75614858012828256769888070842, −7.11362745063964175260237132487, −6.56219048812739275590677879059, −5.28484151066663431212702762802, −4.83464549328545117264910305144, −3.78662983558404878551391607778, −3.39727445270167248855976893761, −1.90343164351988354093423999321, −0.975156422984974655631985791947, 1.51122636176384452130260709305, 2.28956898046998049278417151306, 2.98799035357342695566418471572, 4.46524265923008815291780177896, 4.90046856882448747359728260148, 5.70495714292999236466799728563, 6.43010709491117415246423493199, 7.55973518389088543401967996022, 7.996956592027485703026404130788, 8.610494648800721604879542690694

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