Properties

Label 2-4000-5.3-c0-0-1
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  + (−1.39 − 1.39i)3-s + (0.642 − 0.642i)7-s + 2.90i·9-s − 1.79·21-s + (1.26 + 1.26i)23-s + (2.65 − 2.65i)27-s + 1.61i·29-s + 1.17·41-s + (0.221 + 0.221i)43-s + (−0.221 + 0.221i)47-s + 0.175i·49-s + 1.90·61-s + (1.86 + 1.86i)63-s + (−1 + i)67-s − 3.52i·69-s + ⋯
L(s)  = 1  + (−1.39 − 1.39i)3-s + (0.642 − 0.642i)7-s + 2.90i·9-s − 1.79·21-s + (1.26 + 1.26i)23-s + (2.65 − 2.65i)27-s + 1.61i·29-s + 1.17·41-s + (0.221 + 0.221i)43-s + (−0.221 + 0.221i)47-s + 0.175i·49-s + 1.90·61-s + (1.86 + 1.86i)63-s + (−1 + i)67-s − 3.52i·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

\(L(\frac{1}{2})\) \(\approx\) \(0.8494281454\)
\(L(\frac12)\) \(\approx\) \(0.8494281454\)
\(L(1)\) 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 \( 1 + (1.39 + 1.39i)T + iT^{2} \)
7 \( 1 + (-0.642 + 0.642i)T - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-1.26 - 1.26i)T + iT^{2} \)
29 \( 1 - 1.61iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.17T + T^{2} \)
43 \( 1 + (-0.221 - 0.221i)T + iT^{2} \)
47 \( 1 + (0.221 - 0.221i)T - iT^{2} \)
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 \( 1 + (-0.642 - 0.642i)T + iT^{2} \)
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

−8.207231468710432141380127563183, −7.51316407908959689841889739193, −7.11056725553947279407585714631, −6.46457511320722973930393665498, −5.48766263249432347665791659882, −5.15061634596944888820653100741, −4.20446449558217589288754884532, −2.81215699025586221299169346496, −1.60281600933719855619694639227, −1.01571872114289678813820390500, 0.76714202379998934383389429875, 2.44225631067716770839916983976, 3.58241606877787143565565804641, 4.45774221869678381914768852804, 4.90628253044025022011121201068, 5.66082092459869383260244262741, 6.22797751513219079517244984083, 7.00146652905385253355984182672, 8.168295621737607452760806909314, 8.961784238820055236530000097575

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