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Properties

Label 2-4000-5.3-c0-0-6
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

\(L(\frac{1}{2})\) \(\approx\) \(0.7562467498\)
\(L(\frac12)\) \(\approx\) \(0.7562467498\)
\(L(1)\) 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

−8.284931981271262806103262190644, −7.48098688825115870700739003556, −6.89878047868684388915888753419, −6.06464901266121416766938407234, −5.67871601829204475250348288373, −4.50918643023291850792708552031, −3.90247837241598044999420731036, −2.66940267718612434434992335927, −1.68229271237527879165761840344, −0.46283721215525029424093956826, 1.50060423787936564122638613176, 2.58471226218742250919164149738, 3.71386322009086328585138449273, 4.43673131063969231206036375133, 5.21428702350929557558206117393, 5.81237408284658537514129972172, 6.49734260490206989090093218222, 7.74597899820184922506942652841, 7.88747275125320847945565997013, 9.142443549184207396802143394328

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