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Properties

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

\(L(\frac{1}{2})\) \(\approx\) \(0.3814064260\)
\(L(\frac12)\) \(\approx\) \(0.3814064260\)
\(L(1)\) 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

−9.051863241316194153014577519327, −8.307528241840589688638439980103, −7.29629683455274148301814276069, −6.43472520724932800495126867769, −6.17254922078444473189930811357, −5.38208152447994812193730621052, −4.35328078372044748944789904392, −3.24222008052554387770665793920, −2.81853132084347492985719016947, −1.49317881132663812743318879192, 0.21230574291037070542619742339, 1.79379123833660236090613175825, 2.95615269982111901793949731629, 3.84274249886470454242344384923, 4.40325661955259978700814704877, 5.42687257615480085612451288856, 6.16365240958972830635357714503, 6.92341061842993430285851032829, 7.57614939832784966376014428979, 8.194912876644042680963746445516

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