Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.0706 + 0.997i$
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 + (−0.965 + 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (0.258 + 0.965i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s + i·19-s + (−1.22 − 1.22i)22-s + (0.866 + 0.500i)24-s + (−0.707 + 0.707i)27-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)2-s + (−0.965 + 0.258i)3-s − 1.00i·4-s + (−0.500 + 0.866i)6-s + (−0.707 − 0.707i)8-s + (0.866 − 0.499i)9-s − 1.73i·11-s + (0.258 + 0.965i)12-s − 1.00·16-s + (0.707 − 0.707i)17-s + (0.258 − 0.965i)18-s + i·19-s + (−1.22 − 1.22i)22-s + (0.866 + 0.500i)24-s + (−0.707 + 0.707i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0706 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.0706 + 0.997i$
motivic weight  =  \(0\)
character  :  $\chi_{600} (107, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :0),\ -0.0706 + 0.997i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(0.9545856571\)
\(L(\frac12)\)  \(\approx\)  \(0.9545856571\)
\(L(1)\)   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 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 \)
good7 \( 1 - iT^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 + iT^{2} \)
17 \( 1 + (-0.707 + 0.707i)T - iT^{2} \)
19 \( 1 - iT - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - iT^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 - iT^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + (-1.22 - 1.22i)T + iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + (-0.707 - 0.707i)T + iT^{2} \)
89 \( 1 + 1.73T + T^{2} \)
97 \( 1 + iT^{2} \)
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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

−10.91322260979690947461219863022, −10.06659141973583876358417397174, −9.299362209779989681807293377490, −8.011116415661803315747559785968, −6.58895647036694121778813064625, −5.84010270023975185948784375521, −5.17075549484904348552318256331, −3.98257983847515790569620352270, −3.04905299221340423687937919644, −1.11543928520314058932352548067, 2.13344339935663428547069227099, 3.90200853071741372853193173707, 4.84279187873061813366787269754, 5.55123046536008970087221092949, 6.69908104857988673737869719666, 7.18326940290546465131870257934, 8.098780472366114890942246314833, 9.389317442594265527258093123686, 10.34623537395131639457702074505, 11.32732548520430801547996108605

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