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Properties

Label 2-600-24.11-c1-0-50
Degree $2$
Conductor $600$
Sign $0.445 + 0.895i$
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  + (1.30 − 0.541i)2-s + (−1.13 + 1.30i)3-s + (1.41 − 1.41i)4-s + (−0.778 + 2.32i)6-s − 2.27i·7-s + (1.08 − 2.61i)8-s + (−0.414 − 2.97i)9-s − 4.20i·11-s + (0.239 + 3.45i)12-s + 3.21i·13-s + (−1.23 − 2.97i)14-s − 4i·16-s − 1.53i·17-s + (−2.14 − 3.65i)18-s + 4.82·19-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)2-s + (−0.656 + 0.754i)3-s + (0.707 − 0.707i)4-s + (−0.317 + 0.948i)6-s − 0.859i·7-s + (0.382 − 0.923i)8-s + (−0.138 − 0.990i)9-s − 1.26i·11-s + (0.0692 + 0.997i)12-s + 0.891i·13-s + (−0.328 − 0.794i)14-s − i·16-s − 0.371i·17-s + (−0.506 − 0.862i)18-s + 1.10·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.445 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.445 + 0.895i$
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: \((2,\ 600,\ (\ :1/2),\ 0.445 + 0.895i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72230 - 1.06645i\)
\(L(\frac12)\) \(\approx\) \(1.72230 - 1.06645i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.30 + 0.541i)T \)
3 \( 1 + (1.13 - 1.30i)T \)
5 \( 1 \)
good7 \( 1 + 2.27iT - 7T^{2} \)
11 \( 1 + 4.20iT - 11T^{2} \)
13 \( 1 - 3.21iT - 13T^{2} \)
17 \( 1 + 1.53iT - 17T^{2} \)
19 \( 1 - 4.82T + 19T^{2} \)
23 \( 1 + 1.08T + 23T^{2} \)
29 \( 1 - 1.74T + 29T^{2} \)
31 \( 1 + 6.82iT - 31T^{2} \)
37 \( 1 - 7.76iT - 37T^{2} \)
41 \( 1 - 2.46iT - 41T^{2} \)
43 \( 1 - 8.70T + 43T^{2} \)
47 \( 1 + 1.08T + 47T^{2} \)
53 \( 1 + 11.0T + 53T^{2} \)
59 \( 1 + 4.20iT - 59T^{2} \)
61 \( 1 - 8.48iT - 61T^{2} \)
67 \( 1 + 2.27T + 67T^{2} \)
71 \( 1 + 11.8T + 71T^{2} \)
73 \( 1 + 4.54T + 73T^{2} \)
79 \( 1 - 0.485iT - 79T^{2} \)
83 \( 1 - 6.94iT - 83T^{2} \)
89 \( 1 - 8.40iT - 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−10.71678132335841474566207268300, −9.934080207648075409232593064240, −9.131466042803462028972349671354, −7.60919516785693630193102903943, −6.52308690267103841640840255803, −5.80904597328183481249379998040, −4.76546189496142604088269181384, −3.96438560362147566244491759622, −3.03222557401893425980510585083, −0.971536533219536437717074305587, 1.85620693821595168788650718414, 3.00624950419357672622389322701, 4.59756692226858179581323359884, 5.44823797017475825603002401649, 6.09282541185030463839552094324, 7.21333811432865119035435399972, 7.71415195543690966921726022077, 8.837359448094761821388393109299, 10.20869740795998848056041088257, 11.09750467369372239290295323905

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