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Properties

Label 2-600-120.53-c1-0-36
Degree $2$
Conductor $600$
Sign $0.943 - 0.332i$
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  + (1.21 + 0.722i)2-s + (−1.48 − 0.889i)3-s + (0.956 + 1.75i)4-s + (−1.16 − 2.15i)6-s + (2.09 − 2.09i)7-s + (−0.106 + 2.82i)8-s + (1.41 + 2.64i)9-s − 2.65·11-s + (0.142 − 3.46i)12-s + (4.21 − 4.21i)13-s + (4.05 − 1.03i)14-s + (−2.17 + 3.35i)16-s + (3.84 + 3.84i)17-s + (−0.188 + 4.23i)18-s + 3.15·19-s + ⋯
L(s)  = 1  + (0.859 + 0.510i)2-s + (−0.857 − 0.513i)3-s + (0.478 + 0.878i)4-s + (−0.475 − 0.879i)6-s + (0.790 − 0.790i)7-s + (−0.0377 + 0.999i)8-s + (0.472 + 0.881i)9-s − 0.800·11-s + (0.0411 − 0.999i)12-s + (1.16 − 1.16i)13-s + (1.08 − 0.275i)14-s + (−0.542 + 0.839i)16-s + (0.933 + 0.933i)17-s + (−0.0445 + 0.999i)18-s + 0.724·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.332i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.943 - 0.332i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (293, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.943 - 0.332i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.04214 + 0.349688i\)
\(L(\frac12)\) \(\approx\) \(2.04214 + 0.349688i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.21 - 0.722i)T \)
3 \( 1 + (1.48 + 0.889i)T \)
5 \( 1 \)
good7 \( 1 + (-2.09 + 2.09i)T - 7iT^{2} \)
11 \( 1 + 2.65T + 11T^{2} \)
13 \( 1 + (-4.21 + 4.21i)T - 13iT^{2} \)
17 \( 1 + (-3.84 - 3.84i)T + 17iT^{2} \)
19 \( 1 - 3.15T + 19T^{2} \)
23 \( 1 + (-1.60 + 1.60i)T - 23iT^{2} \)
29 \( 1 - 4.35iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (-4.94 - 4.94i)T + 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (-0.219 + 0.219i)T - 43iT^{2} \)
47 \( 1 + (1.83 + 1.83i)T + 47iT^{2} \)
53 \( 1 + (4.64 + 4.64i)T + 53iT^{2} \)
59 \( 1 - 9.93iT - 59T^{2} \)
61 \( 1 - 11.9iT - 61T^{2} \)
67 \( 1 + (8.80 + 8.80i)T + 67iT^{2} \)
71 \( 1 - 2.89iT - 71T^{2} \)
73 \( 1 + (2.29 + 2.29i)T + 73iT^{2} \)
79 \( 1 + 9.02iT - 79T^{2} \)
83 \( 1 + (9.78 + 9.78i)T + 83iT^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 + (9.26 - 9.26i)T - 97iT^{2} \)
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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

−10.69773991488400197379520583726, −10.48343173516441440450325411130, −8.398777194256961382332947994890, −7.81644479413487836602331724257, −7.11722777933526468870094974856, −5.91619372136630725940863163183, −5.40067412543925945219960463655, −4.37942549911569878327177311867, −3.13766315816965008371335401699, −1.32152270736160565026666097059, 1.34841341987570959009741113746, 2.91361776630124424283330658364, 4.17790930372629512748060907011, 5.08368995023020904904633466929, 5.69932982910056834108579733889, 6.61945727500506997288456999512, 7.88621889510037745450799330422, 9.296788848662018569760179937474, 9.864467191926192433991483296125, 11.14844945805387480814944681041

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