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  :  $\chi_{3600} (451, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 3600,\ (\ :0),\ -0.382 - 0.923i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.2434917459\)
\(L(\frac12)\)  \(\approx\)  \(0.2434917459\)
\(L(1)\)   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

−8.807908588071774236629396179495, −8.476538488618976638541710143482, −7.71909858512125339816937232839, −6.78853433220247096546344185676, −6.24597591718127930628764576941, −4.98217247665878702418684643061, −4.14603874487716110791227026129, −3.44094805437287590450927038824, −2.30813523539259161066549742251, −1.61437694517079147150296135734, 0.16671518267732937658998879965, 1.81600961662093426441045057405, 2.60723553073464479934983299064, 4.24353879066118819682560675171, 4.61350983078824726940189166123, 5.87879829589252799161627287048, 6.26604891605964461487472289660, 7.07475716280327681707089289740, 7.81080650170841406206213795148, 8.479955180473945963497386086152

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