Properties

Degree 2
Conductor $ 2^{3} \cdot 3 \cdot 5^{2} $
Sign $-0.563 + 0.825i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.450 − 1.34i)2-s i·3-s + (−1.59 + 1.20i)4-s + (−1.34 + 0.450i)6-s + 2.64·7-s + (2.33 + 1.59i)8-s − 9-s + 1.51i·11-s + (1.20 + 1.59i)12-s − 3.87i·13-s + (−1.18 − 3.54i)14-s + (1.08 − 3.84i)16-s + 3.31·17-s + (0.450 + 1.34i)18-s − 7.08i·19-s + ⋯
L(s)  = 1  + (−0.318 − 0.947i)2-s − 0.577i·3-s + (−0.797 + 0.603i)4-s + (−0.547 + 0.183i)6-s + 0.998·7-s + (0.825 + 0.563i)8-s − 0.333·9-s + 0.456i·11-s + (0.348 + 0.460i)12-s − 1.07i·13-s + (−0.317 − 0.946i)14-s + (0.271 − 0.962i)16-s + 0.803·17-s + (0.106 + 0.315i)18-s − 1.62i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
\( \varepsilon \)  =  $-0.563 + 0.825i$
motivic weight  =  \(1\)
character  :  $\chi_{600} (301, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 600,\ (\ :1/2),\ -0.563 + 0.825i)\)
\(L(1)\)  \(\approx\)  \(0.580760 - 1.09968i\)
\(L(\frac12)\)  \(\approx\)  \(0.580760 - 1.09968i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.450 + 1.34i)T \)
3 \( 1 + iT \)
5 \( 1 \)
good7 \( 1 - 2.64T + 7T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 + 3.87iT - 13T^{2} \)
17 \( 1 - 3.31T + 17T^{2} \)
19 \( 1 + 7.08iT - 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 - 2.18iT - 29T^{2} \)
31 \( 1 + 7.36T + 31T^{2} \)
37 \( 1 + 7.87iT - 37T^{2} \)
41 \( 1 - 8.72T + 41T^{2} \)
43 \( 1 + 1.01iT - 43T^{2} \)
47 \( 1 + 7.08T + 47T^{2} \)
53 \( 1 + 4.50iT - 53T^{2} \)
59 \( 1 - 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 + 1.01iT - 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 - 15.5T + 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 + 7.74iT - 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.1T + 97T^{2} \)
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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

−10.69203142614528882343799538061, −9.454964880441780168858609357331, −8.724597621474095274475171309209, −7.74985373691166743122817674619, −7.21048030403424997239023831354, −5.48250708016272044271730996906, −4.73064519309181136270615358471, −3.29917975632862668681258457054, −2.18850875419718800536783374044, −0.888008447718660803343565561406, 1.51681058281702573856635576134, 3.64297498736320779985968351396, 4.66439313985363198200895337044, 5.49325729170002586813723452417, 6.41392736614325215224995099872, 7.62737890401704711843604772332, 8.239397035737924239656284424862, 9.145416366014996883600986693796, 9.876962666735909855418351663180, 10.83753184063489894435602937065

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