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Properties

Label 2-4000-125.113-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.303 + 0.952i$
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  + (0.368 − 0.929i)5-s + (−0.684 − 0.728i)9-s + (−0.188 − 0.00591i)13-s + (−0.0613 − 0.0137i)17-s + (−0.728 − 0.684i)25-s + (0.450 − 1.75i)29-s + (1.48 + 0.535i)37-s + (−1.67 + 0.211i)41-s + (−0.929 + 0.368i)45-s + (−0.587 − 0.809i)49-s + (0.162 − 1.71i)53-s + (−0.730 − 0.0922i)61-s + (−0.0747 + 0.172i)65-s + (0.843 − 1.24i)73-s + (−0.0627 + 0.998i)81-s + ⋯
L(s)  = 1  + (0.368 − 0.929i)5-s + (−0.684 − 0.728i)9-s + (−0.188 − 0.00591i)13-s + (−0.0613 − 0.0137i)17-s + (−0.728 − 0.684i)25-s + (0.450 − 1.75i)29-s + (1.48 + 0.535i)37-s + (−1.67 + 0.211i)41-s + (−0.929 + 0.368i)45-s + (−0.587 − 0.809i)49-s + (0.162 − 1.71i)53-s + (−0.730 − 0.0922i)61-s + (−0.0747 + 0.172i)65-s + (0.843 − 1.24i)73-s + (−0.0627 + 0.998i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.303 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.303 + 0.952i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (2113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.303 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.076174986\)
\(L(\frac12)\) \(\approx\) \(1.076174986\)
\(L(1)\) 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 \( 1 + (-0.368 + 0.929i)T \)
good3 \( 1 + (0.684 + 0.728i)T^{2} \)
7 \( 1 + (0.587 + 0.809i)T^{2} \)
11 \( 1 + (0.535 - 0.844i)T^{2} \)
13 \( 1 + (0.188 + 0.00591i)T + (0.998 + 0.0627i)T^{2} \)
17 \( 1 + (0.0613 + 0.0137i)T + (0.904 + 0.425i)T^{2} \)
19 \( 1 + (-0.728 - 0.684i)T^{2} \)
23 \( 1 + (-0.248 + 0.968i)T^{2} \)
29 \( 1 + (-0.450 + 1.75i)T + (-0.876 - 0.481i)T^{2} \)
31 \( 1 + (-0.425 + 0.904i)T^{2} \)
37 \( 1 + (-1.48 - 0.535i)T + (0.770 + 0.637i)T^{2} \)
41 \( 1 + (1.67 - 0.211i)T + (0.968 - 0.248i)T^{2} \)
43 \( 1 + (0.951 + 0.309i)T^{2} \)
47 \( 1 + (0.125 + 0.992i)T^{2} \)
53 \( 1 + (-0.162 + 1.71i)T + (-0.982 - 0.187i)T^{2} \)
59 \( 1 + (0.929 + 0.368i)T^{2} \)
61 \( 1 + (0.730 + 0.0922i)T + (0.968 + 0.248i)T^{2} \)
67 \( 1 + (-0.481 - 0.876i)T^{2} \)
71 \( 1 + (-0.992 + 0.125i)T^{2} \)
73 \( 1 + (-0.843 + 1.24i)T + (-0.368 - 0.929i)T^{2} \)
79 \( 1 + (-0.728 + 0.684i)T^{2} \)
83 \( 1 + (-0.684 + 0.728i)T^{2} \)
89 \( 1 + (0.946 - 0.180i)T + (0.929 - 0.368i)T^{2} \)
97 \( 1 + (1.42 - 0.842i)T + (0.481 - 0.876i)T^{2} \)
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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

−8.248230210683192228915505188357, −8.065209643405111352306898393094, −6.73756237392658790586021038590, −6.20167359527409834467993238239, −5.42286862566325120877861283505, −4.70072117071631637002763539211, −3.86316544373755689513486221666, −2.86379152273808265814118016654, −1.85514255835504496449787674608, −0.58315218715234647339268454366, 1.60929607337890690165863046269, 2.65322076608402881042698091541, 3.17571845897449453356353138024, 4.32800356264551850419370505776, 5.24811140703055185997373101442, 5.87716344352615467659267972353, 6.69089771498041998000243529131, 7.33649004444747284339150477891, 8.067861766054462867047872272419, 8.826578802159391487918658110514

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