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Properties

Label 2-4000-5.3-c0-0-4
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  + (0.642 + 0.642i)3-s + (−0.221 + 0.221i)7-s − 0.175i·9-s − 0.284·21-s + (1.39 + 1.39i)23-s + (0.754 − 0.754i)27-s − 0.618i·29-s + 1.90·41-s + (1.26 + 1.26i)43-s + (−1.26 + 1.26i)47-s + 0.902i·49-s − 1.17·61-s + (0.0388 + 0.0388i)63-s + (1 − i)67-s + 1.79i·69-s + ⋯
L(s)  = 1  + (0.642 + 0.642i)3-s + (−0.221 + 0.221i)7-s − 0.175i·9-s − 0.284·21-s + (1.39 + 1.39i)23-s + (0.754 − 0.754i)27-s − 0.618i·29-s + 1.90·41-s + (1.26 + 1.26i)43-s + (−1.26 + 1.26i)47-s + 0.902i·49-s − 1.17·61-s + (0.0388 + 0.0388i)63-s + (1 − i)67-s + 1.79i·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

\(L(\frac{1}{2})\) \(\approx\) \(1.633952288\)
\(L(\frac12)\) \(\approx\) \(1.633952288\)
\(L(1)\) 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 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
7 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.39 - 1.39i)T + iT^{2} \)
29 \( 1 + 0.618iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.90T + T^{2} \)
43 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
47 \( 1 + (1.26 - 1.26i)T - iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.17T + T^{2} \)
67 \( 1 + (-1 + i)T - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + iT^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (0.221 + 0.221i)T + iT^{2} \)
89 \( 1 + 1.61iT - T^{2} \)
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

−8.963582057730838117657885801932, −7.976084480483598534631227987987, −7.45893461931368195777895935112, −6.41548332253412383434624464887, −5.81677437182062153528388304966, −4.78858977560207634701722575565, −4.12400966824323883224249478523, −3.19570448806381838840968905931, −2.67830536306665068401788467730, −1.24370279331034385931454457323, 1.00612012080376843422467819754, 2.21551729182357298833747047106, 2.85853008622704702695439656293, 3.83644680583641933712709119530, 4.78582063171766623249038818079, 5.54564101967903129121951475471, 6.64125138139568672201001514230, 7.06645050823975403785880737516, 7.79149427343800422568427464988, 8.559708078627678259924138133096

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