Properties

Degree 2
Conductor $ 2^{4} \cdot 3^{2} \cdot 5^{2} $
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

\( d \)  =  \(2\)
\( N \)  =  \(3600\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.382 + 0.923i$
motivic weight  =  \(0\)
character  :  $\chi_{3600} (2251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3600,\ (\ :0),\ -0.382 + 0.923i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.788782711\)
\(L(\frac12)\)  \(\approx\)  \(1.788782711\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 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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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

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

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