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  =  \((2,\ 600,\ (\ :1/2),\ 0.634 - 0.773i)\)
\(L(1)\)  \(\approx\)  \(0.527029 + 0.249334i\)
\(L(\frac12)\)  \(\approx\)  \(0.527029 + 0.249334i\)
\(L(\frac{3}{2})\)   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

−10.69231361497728749034256576335, −9.697614190469730446361806798678, −9.208162484958449514264695903082, −8.254268878368293960490522866126, −7.13083069633664159001346794529, −6.65672533816700840090504311701, −5.69671330409677847557970800821, −3.69871513588869086560805485087, −2.80989517177547999257009869207, −1.31237399614592677664554219641, 0.47502047449645963913180043947, 2.68012783790873851594149227982, 3.54272184781054760414249278600, 5.31466687036696724397622874714, 6.07492156558734245450330417957, 7.09255284334073892036095905749, 7.952709504415778200390326834130, 9.024087839831539871846265937985, 9.723255845577607991878836831596, 10.23730653554267955195949071697

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