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Properties

Label 2-40e2-100.11-c0-0-1
Degree $2$
Conductor $1600$
Sign $-0.187 + 0.982i$
Analytic cond. $0.798504$
Root an. cond. $0.893590$
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  + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)9-s + (0.5 + 0.363i)13-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (−1.30 − 0.951i)37-s + (−0.5 − 0.363i)41-s + (−0.309 + 0.951i)45-s + 49-s + (−0.190 + 0.587i)53-s + (0.5 − 0.363i)61-s + (0.190 − 0.587i)65-s + (−0.5 + 0.363i)73-s + (0.309 + 0.951i)81-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)5-s + (−0.809 − 0.587i)9-s + (0.5 + 0.363i)13-s + (−0.5 − 1.53i)17-s + (−0.809 + 0.587i)25-s + (0.5 − 1.53i)29-s + (−1.30 − 0.951i)37-s + (−0.5 − 0.363i)41-s + (−0.309 + 0.951i)45-s + 49-s + (−0.190 + 0.587i)53-s + (0.5 − 0.363i)61-s + (0.190 − 0.587i)65-s + (−0.5 + 0.363i)73-s + (0.309 + 0.951i)81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.187 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1600\)    =    \(2^{6} \cdot 5^{2}\)
Sign: $-0.187 + 0.982i$
Analytic conductor: \(0.798504\)
Root analytic conductor: \(0.893590\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1600} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1600,\ (\ :0),\ -0.187 + 0.982i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8391105226\)
\(L(\frac12)\) \(\approx\) \(0.8391105226\)
\(L(1)\) 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 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 + 0.951i)T^{2} \)
13 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
17 \( 1 + (0.5 + 1.53i)T + (-0.809 + 0.587i)T^{2} \)
19 \( 1 + (0.809 - 0.587i)T^{2} \)
23 \( 1 + (-0.309 + 0.951i)T^{2} \)
29 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.809 - 0.587i)T^{2} \)
37 \( 1 + (1.30 + 0.951i)T + (0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 + 0.587i)T^{2} \)
53 \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \)
59 \( 1 + (-0.309 - 0.951i)T^{2} \)
61 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 + (0.809 - 0.587i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
79 \( 1 + (0.809 + 0.587i)T^{2} \)
83 \( 1 + (0.809 - 0.587i)T^{2} \)
89 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
97 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
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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

−9.060530546209481063756938441076, −8.849292540963149531341311797377, −7.88585154815558867769082933207, −7.00371766543096091699328867620, −6.04995159932846715639649453724, −5.24443509450568258219331981418, −4.37528734040920828418377688983, −3.46109081550391221025307731017, −2.24673950601836737448916783596, −0.65014870029216572411316128899, 1.83983109858557752187143826474, 3.01162637887682082632787829172, 3.71058196575893925915097754418, 4.90580606702507158354179508218, 5.88923826015952475583707126015, 6.57696057727015355967606272153, 7.42126686632240611254181321579, 8.417282979627949200631293199865, 8.674595640950674101332078338257, 10.12576251603664497746291482382

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