↑

Properties

Label 2-4000-125.112-c0-0-0
Degree $2$
Conductor $4000$
Sign $-0.592 - 0.805i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.998 − 0.0627i)5-s + (−0.125 − 0.992i)9-s + (−0.486 + 0.627i)13-s + (−1.57 + 0.148i)17-s + (0.992 + 0.125i)25-s + (1.65 + 1.05i)29-s + (−1.58 + 1.07i)37-s + (−1.35 − 0.742i)41-s + (0.0627 + 0.998i)45-s + (0.587 + 0.809i)49-s + (−0.775 + 1.79i)53-s + (−1.74 + 0.961i)61-s + (0.525 − 0.595i)65-s + (1.01 − 0.0319i)73-s + (−0.968 + 0.248i)81-s + ⋯
L(s)  = 1  + (−0.998 − 0.0627i)5-s + (−0.125 − 0.992i)9-s + (−0.486 + 0.627i)13-s + (−1.57 + 0.148i)17-s + (0.992 + 0.125i)25-s + (1.65 + 1.05i)29-s + (−1.58 + 1.07i)37-s + (−1.35 − 0.742i)41-s + (0.0627 + 0.998i)45-s + (0.587 + 0.809i)49-s + (−0.775 + 1.79i)53-s + (−1.74 + 0.961i)61-s + (0.525 − 0.595i)65-s + (1.01 − 0.0319i)73-s + (−0.968 + 0.248i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.592 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4000\)    =    \(2^{5} \cdot 5^{3}\)
Sign: $-0.592 - 0.805i$
Analytic conductor: \(1.99626\)
Root analytic conductor: \(1.41289\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4000} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4000,\ (\ :0),\ -0.592 - 0.805i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3520423598\)
\(L(\frac12)\) \(\approx\) \(0.3520423598\)
\(L(1)\) 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 \( 1 + (0.998 + 0.0627i)T \)
good3 \( 1 + (0.125 + 0.992i)T^{2} \)
7 \( 1 + (-0.587 - 0.809i)T^{2} \)
11 \( 1 + (-0.637 - 0.770i)T^{2} \)
13 \( 1 + (0.486 - 0.627i)T + (-0.248 - 0.968i)T^{2} \)
17 \( 1 + (1.57 - 0.148i)T + (0.982 - 0.187i)T^{2} \)
19 \( 1 + (0.992 + 0.125i)T^{2} \)
23 \( 1 + (-0.844 - 0.535i)T^{2} \)
29 \( 1 + (-1.65 - 1.05i)T + (0.425 + 0.904i)T^{2} \)
31 \( 1 + (-0.187 - 0.982i)T^{2} \)
37 \( 1 + (1.58 - 1.07i)T + (0.368 - 0.929i)T^{2} \)
41 \( 1 + (1.35 + 0.742i)T + (0.535 + 0.844i)T^{2} \)
43 \( 1 + (-0.951 - 0.309i)T^{2} \)
47 \( 1 + (-0.481 + 0.876i)T^{2} \)
53 \( 1 + (0.775 - 1.79i)T + (-0.684 - 0.728i)T^{2} \)
59 \( 1 + (-0.0627 + 0.998i)T^{2} \)
61 \( 1 + (1.74 - 0.961i)T + (0.535 - 0.844i)T^{2} \)
67 \( 1 + (-0.904 - 0.425i)T^{2} \)
71 \( 1 + (0.876 + 0.481i)T^{2} \)
73 \( 1 + (-1.01 + 0.0319i)T + (0.998 - 0.0627i)T^{2} \)
79 \( 1 + (0.992 - 0.125i)T^{2} \)
83 \( 1 + (-0.125 + 0.992i)T^{2} \)
89 \( 1 + (1.23 - 1.31i)T + (-0.0627 - 0.998i)T^{2} \)
97 \( 1 + (-0.0613 + 0.0137i)T + (0.904 - 0.425i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.737741721106932966419080238852, −8.396957661082891940117683959043, −7.17791630859979823841961134115, −6.87136974645224809439573181262, −6.12373257190280092567345674586, −4.85644051728215680771114966214, −4.42366824292623955314740557529, −3.50476452920446201471605306821, −2.74001902876682082282184823193, −1.37506656756952900579980608013, 0.19684317310263933227371416600, 1.98091092929615412590089834512, 2.84012970985911837625513693756, 3.76961480877980016277782134691, 4.77300378586493367144319566603, 5.03839980039416211450818637901, 6.32468902743415703086753460849, 6.98074834692967785963640315043, 7.71413198170724737767953506550, 8.353451751564676672206963979616

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