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Properties

Label 2-4000-125.22-c0-0-0
Degree $2$
Conductor $4000$
Sign $0.999 + 0.0439i$
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  + (−0.904 + 0.425i)5-s + (0.770 − 0.637i)9-s + (0.555 + 0.0525i)13-s + (0.148 + 0.115i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−0.404 − 0.683i)37-s + (0.233 − 0.0922i)41-s + (−0.425 + 0.904i)45-s + (0.951 − 0.309i)49-s + (0.0175 − 0.0603i)53-s + (1.68 + 0.666i)61-s + (−0.524 + 0.189i)65-s + (1.09 + 0.245i)73-s + (0.187 − 0.982i)81-s + ⋯
L(s)  = 1  + (−0.904 + 0.425i)5-s + (0.770 − 0.637i)9-s + (0.555 + 0.0525i)13-s + (0.148 + 0.115i)17-s + (0.637 − 0.770i)25-s + (−0.340 + 0.362i)29-s + (−0.404 − 0.683i)37-s + (0.233 − 0.0922i)41-s + (−0.425 + 0.904i)45-s + (0.951 − 0.309i)49-s + (0.0175 − 0.0603i)53-s + (1.68 + 0.666i)61-s + (−0.524 + 0.189i)65-s + (1.09 + 0.245i)73-s + (0.187 − 0.982i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $0.999 + 0.0439i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (897, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ 0.999 + 0.0439i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.187606248\)
\(L(\frac12)\) \(\approx\) \(1.187606248\)
\(L(1)\) 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 \( 1 + (0.904 - 0.425i)T \)
good3 \( 1 + (-0.770 + 0.637i)T^{2} \)
7 \( 1 + (-0.951 + 0.309i)T^{2} \)
11 \( 1 + (-0.992 - 0.125i)T^{2} \)
13 \( 1 + (-0.555 - 0.0525i)T + (0.982 + 0.187i)T^{2} \)
17 \( 1 + (-0.148 - 0.115i)T + (0.248 + 0.968i)T^{2} \)
19 \( 1 + (0.637 - 0.770i)T^{2} \)
23 \( 1 + (0.684 - 0.728i)T^{2} \)
29 \( 1 + (0.340 - 0.362i)T + (-0.0627 - 0.998i)T^{2} \)
31 \( 1 + (0.968 - 0.248i)T^{2} \)
37 \( 1 + (0.404 + 0.683i)T + (-0.481 + 0.876i)T^{2} \)
41 \( 1 + (-0.233 + 0.0922i)T + (0.728 - 0.684i)T^{2} \)
43 \( 1 + (0.587 + 0.809i)T^{2} \)
47 \( 1 + (-0.368 - 0.929i)T^{2} \)
53 \( 1 + (-0.0175 + 0.0603i)T + (-0.844 - 0.535i)T^{2} \)
59 \( 1 + (0.425 + 0.904i)T^{2} \)
61 \( 1 + (-1.68 - 0.666i)T + (0.728 + 0.684i)T^{2} \)
67 \( 1 + (0.998 + 0.0627i)T^{2} \)
71 \( 1 + (-0.929 + 0.368i)T^{2} \)
73 \( 1 + (-1.09 - 0.245i)T + (0.904 + 0.425i)T^{2} \)
79 \( 1 + (0.637 + 0.770i)T^{2} \)
83 \( 1 + (0.770 + 0.637i)T^{2} \)
89 \( 1 + (-1.68 + 1.06i)T + (0.425 - 0.904i)T^{2} \)
97 \( 1 + (0.0137 + 0.436i)T + (-0.998 + 0.0627i)T^{2} \)
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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

−8.635657497459925266579849385477, −7.78738895912170723968354377010, −7.16338554340347677225887969595, −6.58944944740045599805596410142, −5.73593379393305119391555410804, −4.71998713997281095182706370104, −3.84406610784438934985087056451, −3.46875807111809548420072972950, −2.23034026282355052912704346301, −0.915823152696376132682575643525, 0.999606620217287016226190991988, 2.14038891335363813846497348116, 3.37233949070061132517144875034, 4.07761999054164066234168643214, 4.80429926087061022184956794349, 5.52244082741293673388484431838, 6.56820303495944578365519835708, 7.29089800720330840724243054872, 7.931636386587508908325756121990, 8.475801249500912044976708730325

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