Properties

Label 2-600-24.11-c1-0-22
Degree $2$
Conductor $600$
Sign $-0.214 - 0.976i$
Analytic cond. $4.79102$
Root an. cond. $2.18884$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.13 + 0.842i)2-s + (1.71 + 0.218i)3-s + (0.581 − 1.91i)4-s + (−2.13 + 1.19i)6-s + 3.64i·7-s + (0.949 + 2.66i)8-s + (2.90 + 0.750i)9-s + 5.07i·11-s + (1.41 − 3.16i)12-s − 1.70i·13-s + (−3.06 − 4.14i)14-s + (−3.32 − 2.22i)16-s − 4.08i·17-s + (−3.93 + 1.59i)18-s − 1.26·19-s + ⋯
L(s)  = 1  + (−0.803 + 0.595i)2-s + (0.992 + 0.126i)3-s + (0.290 − 0.956i)4-s + (−0.872 + 0.489i)6-s + 1.37i·7-s + (0.335 + 0.941i)8-s + (0.968 + 0.250i)9-s + 1.52i·11-s + (0.409 − 0.912i)12-s − 0.473i·13-s + (−0.820 − 1.10i)14-s + (−0.830 − 0.556i)16-s − 0.989i·17-s + (−0.926 + 0.375i)18-s − 0.290·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.214 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.214 - 0.976i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.214 - 0.976i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848446 + 1.05483i\)
\(L(\frac12)\) \(\approx\) \(0.848446 + 1.05483i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.13 - 0.842i)T \)
3 \( 1 + (-1.71 - 0.218i)T \)
5 \( 1 \)
good7 \( 1 - 3.64iT - 7T^{2} \)
11 \( 1 - 5.07iT - 11T^{2} \)
13 \( 1 + 1.70iT - 13T^{2} \)
17 \( 1 + 4.08iT - 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 + 4.70T + 23T^{2} \)
29 \( 1 - 1.06T + 29T^{2} \)
31 \( 1 - 4.86iT - 31T^{2} \)
37 \( 1 - 7.56iT - 37T^{2} \)
41 \( 1 + 1.50iT - 41T^{2} \)
43 \( 1 + 3.43T + 43T^{2} \)
47 \( 1 - 10.9T + 47T^{2} \)
53 \( 1 - 8.87T + 53T^{2} \)
59 \( 1 + 0.788iT - 59T^{2} \)
61 \( 1 - 0.627iT - 61T^{2} \)
67 \( 1 + 4.18T + 67T^{2} \)
71 \( 1 - 6.21T + 71T^{2} \)
73 \( 1 - 4.21T + 73T^{2} \)
79 \( 1 + 0.992iT - 79T^{2} \)
83 \( 1 + 7.72iT - 83T^{2} \)
89 \( 1 + 11.5iT - 89T^{2} \)
97 \( 1 - 7.40T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.39198864674682378679897013531, −9.806694087357041867525765072090, −9.059628421544978933071269678283, −8.407903130089243481284597045063, −7.52998746442680266964796998890, −6.73396349309192820842824126160, −5.47934232353476813126385330331, −4.56213891335060551947068830490, −2.74123700692087512299992418529, −1.88145154013887066351264206974, 0.909053892361819022960348323984, 2.28704652960574030772927279198, 3.70788260878489742516689836247, 4.03054001071393797792990807888, 6.21881033400318296706533021760, 7.22484542776938898123498743446, 8.013403966512817820105244000435, 8.625536113968931070083503170420, 9.527330072170810924538303936307, 10.45153958203214267856634898581

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