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Properties

Label 2-60e2-100.91-c0-0-1
Degree $2$
Conductor $3600$
Sign $0.992 + 0.125i$
Analytic cond. $1.79663$
Root an. cond. $1.34038$
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  + (0.951 + 0.309i)5-s + (0.5 − 0.363i)13-s + (0.363 − 1.11i)17-s + (0.809 + 0.587i)25-s + (−0.363 − 1.11i)29-s + (1.30 − 0.951i)37-s + (−1.53 + 1.11i)41-s + 49-s + (0.587 + 1.80i)53-s + (−0.5 − 0.363i)61-s + (0.587 − 0.190i)65-s + (0.5 + 0.363i)73-s + (0.690 − 0.951i)85-s + (−0.951 − 0.690i)89-s + (0.5 + 1.53i)97-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)5-s + (0.5 − 0.363i)13-s + (0.363 − 1.11i)17-s + (0.809 + 0.587i)25-s + (−0.363 − 1.11i)29-s + (1.30 − 0.951i)37-s + (−1.53 + 1.11i)41-s + 49-s + (0.587 + 1.80i)53-s + (−0.5 − 0.363i)61-s + (0.587 − 0.190i)65-s + (0.5 + 0.363i)73-s + (0.690 − 0.951i)85-s + (−0.951 − 0.690i)89-s + (0.5 + 1.53i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.125i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $0.992 + 0.125i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ 0.992 + 0.125i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.594064172\)
\(L(\frac12)\) \(\approx\) \(1.594064172\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.951 - 0.309i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 + (-0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.5 + 0.363i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (-0.363 + 1.11i)T + (-0.809 - 0.587i)T^{2} \)
19 \( 1 + (0.809 + 0.587i)T^{2} \)
23 \( 1 + (-0.309 - 0.951i)T^{2} \)
29 \( 1 + (0.363 + 1.11i)T + (-0.809 + 0.587i)T^{2} \)
31 \( 1 + (0.809 + 0.587i)T^{2} \)
37 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
41 \( 1 + (1.53 - 1.11i)T + (0.309 - 0.951i)T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.587 - 1.80i)T + (-0.809 + 0.587i)T^{2} \)
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 \( 1 + (0.951 + 0.690i)T + (0.309 + 0.951i)T^{2} \)
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

−8.846425148562110702671860661194, −7.899422566098455873826317856392, −7.25324316265774580586396200633, −6.37149469470094542429778690296, −5.78383123196327227248882909851, −5.07773739855096311175060755054, −4.08819744614148339583495377159, −3.02422925508412039151182589695, −2.33358185009805151896897495565, −1.11163318493477804693663045786, 1.30301573505290493334194794509, 2.09305214864547005799939317100, 3.25419083630248390471470741724, 4.12413853528229830089024336842, 5.10799914583940680299134860673, 5.73975904881608301088607925188, 6.46419308067215950710019707813, 7.12969423379416463816538891214, 8.265518034654307971162576259352, 8.668696506000398616925272175740

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