Properties

Label 2-60e2-60.23-c0-0-1
Degree $2$
Conductor $3600$
Sign $-0.391 + 0.920i$
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  + (−1 − i)13-s + (−1.41 − 1.41i)17-s − 1.41·29-s + (1 − i)37-s − 1.41i·41-s i·49-s + (−1.41 + 1.41i)53-s + (1 + i)73-s + 1.41·89-s + (1 − i)97-s − 1.41i·101-s + ⋯
L(s)  = 1  + (−1 − i)13-s + (−1.41 − 1.41i)17-s − 1.41·29-s + (1 − i)37-s − 1.41i·41-s i·49-s + (−1.41 + 1.41i)53-s + (1 + i)73-s + 1.41·89-s + (1 − i)97-s − 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.391 + 0.920i$
Analytic conductor: \(1.79663\)
Root analytic conductor: \(1.34038\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3600} (143, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.391 + 0.920i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7407791707\)
\(L(\frac12)\) \(\approx\) \(0.7407791707\)
\(L(1)\) 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 \( 1 \)
good7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 + i)T + iT^{2} \)
17 \( 1 + (1.41 + 1.41i)T + iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 + i)T - iT^{2} \)
41 \( 1 + 1.41iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 - i)T + iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (-1 + i)T - iT^{2} \)
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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

−8.567248892908303044708224629718, −7.46839342528560548135297605897, −7.32181432205187540716255600752, −6.25443319809882618780658313562, −5.39124956076113367634690634621, −4.78952659669949514086963591717, −3.86226711840103339051260172007, −2.79469070822809598425280377879, −2.10734300108169582526941406054, −0.39850318103113549641196065899, 1.65719099457495338257020987190, 2.41360974141554376186838932514, 3.60535296727699148717555715829, 4.43822268827050100980762420462, 5.01615814203792437299528726007, 6.29596244471888632877575886245, 6.50876609373685622225389472634, 7.58603639358810626738074282252, 8.142141148309820052391565959894, 9.103922071014337572811379018402

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