Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $0.634 + 0.773i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.37 − 0.321i)2-s i·3-s + (1.79 − 0.884i)4-s + (−0.321 − 1.37i)6-s + 4.05·7-s + (2.18 − 1.79i)8-s − 9-s + 0.985i·11-s + (−0.884 − 1.79i)12-s + 4.94i·13-s + (5.58 − 1.30i)14-s + (2.43 − 3.17i)16-s − 4.52·17-s + (−1.37 + 0.321i)18-s − 2.60i·19-s + ⋯
L(s)  = 1  + (0.973 − 0.227i)2-s − 0.577i·3-s + (0.896 − 0.442i)4-s + (−0.131 − 0.562i)6-s + 1.53·7-s + (0.773 − 0.634i)8-s − 0.333·9-s + 0.297i·11-s + (−0.255 − 0.517i)12-s + 1.37i·13-s + (1.49 − 0.348i)14-s + (0.608 − 0.793i)16-s − 1.09·17-s + (−0.324 + 0.0756i)18-s − 0.597i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.634 + 0.773i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $0.634 + 0.773i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (301, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ 0.634 + 0.773i)\)
\(L(1)\)  \(\approx\)  \(2.69984 - 1.27728i\)
\(L(\frac12)\)  \(\approx\)  \(2.69984 - 1.27728i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.37 + 0.321i)T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - 4.05T + 7T^{2} \)
11 \( 1 - 0.985iT - 11T^{2} \)
13 \( 1 - 4.94iT - 13T^{2} \)
17 \( 1 + 4.52T + 17T^{2} \)
19 \( 1 + 2.60iT - 19T^{2} \)
23 \( 1 + 3.53T + 23T^{2} \)
29 \( 1 + 7.59iT - 29T^{2} \)
31 \( 1 + 3.28T + 31T^{2} \)
37 \( 1 - 0.945iT - 37T^{2} \)
41 \( 1 - 0.568T + 41T^{2} \)
43 \( 1 - 8.45iT - 43T^{2} \)
47 \( 1 + 2.60T + 47T^{2} \)
53 \( 1 - 0.229iT - 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 + 11.0iT - 61T^{2} \)
67 \( 1 - 8.45iT - 67T^{2} \)
71 \( 1 - 1.43T + 71T^{2} \)
73 \( 1 + 11.9T + 73T^{2} \)
79 \( 1 + 3.28T + 79T^{2} \)
83 \( 1 - 9.89iT - 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 - 3.23T + 97T^{2} \)
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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

−11.17556389033025547471859246489, −9.806527526196937098883729756714, −8.649414995010903821178257844332, −7.66329615607892322994355243518, −6.85797949351562509081620825108, −5.94393994988068007875665690663, −4.72453795791167480090982330025, −4.21237139096309435298353476617, −2.39050774276089377667442248232, −1.63079743699465119218794675979, 1.90867234337373783480437879351, 3.29280473848373169082831061709, 4.35533038904668262351051478384, 5.19423053894751169750780622586, 5.86307802679649085233981382833, 7.22477772402297381302696789290, 8.103795910969753533690696681547, 8.744620434809704144504353511062, 10.35949617081439545864297827502, 10.86900679388285251613165369008

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