Properties

Degree $2$
Conductor $3600$
Sign $-0.382 - 0.923i$
Motivic weight $0$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s − 1.00·16-s + 1.41·17-s + (−1 + i)19-s + 1.41·23-s + 2i·31-s + (−0.707 − 0.707i)32-s + (1.00 + 1.00i)34-s − 1.41·38-s + (1.00 + 1.00i)46-s + 1.41i·47-s − 49-s + (−1 − i)61-s + (−1.41 + 1.41i)62-s + ⋯
L(s)  = 1  + (0.707 + 0.707i)2-s + 1.00i·4-s + (−0.707 + 0.707i)8-s − 1.00·16-s + 1.41·17-s + (−1 + i)19-s + 1.41·23-s + 2i·31-s + (−0.707 − 0.707i)32-s + (1.00 + 1.00i)34-s − 1.41·38-s + (1.00 + 1.00i)46-s + 1.41i·47-s − 49-s + (−1 − i)61-s + (−1.41 + 1.41i)62-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
Sign: $-0.382 - 0.923i$
Motivic weight: \(0\)
Character: $\chi_{3600} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3600,\ (\ :0),\ -0.382 - 0.923i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.788782711\)
\(L(\frac12)\) \(\approx\) \(1.788782711\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + T^{2} \)
11 \( 1 + iT^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 - 1.41T + T^{2} \)
19 \( 1 + (1 - i)T - iT^{2} \)
23 \( 1 - 1.41T + T^{2} \)
29 \( 1 + iT^{2} \)
31 \( 1 - 2iT - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 - 1.41iT - T^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + iT^{2} \)
61 \( 1 + (1 + i)T + iT^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-1.41 + 1.41i)T - iT^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.739548244820448964306553226188, −8.064195037630132165942152590804, −7.44391627976619309053602629073, −6.63365072542530337417233310140, −6.00477529917657942880727165907, −5.17550548795675326349954246620, −4.57505655808519907944404620231, −3.49023799246816021736920406591, −2.98954593424674399346939061585, −1.56873612633801131118794968377, 0.864490264831248719013907376912, 2.11342649451809806666003523781, 2.96059460854360747427595305849, 3.78500489946826328544568450396, 4.64682768377721411346024987088, 5.33182694746686031333768168516, 6.10727422809373159310536179858, 6.86213226580840202940040756303, 7.69219125961290442065247765548, 8.677544458126776470747749793296

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