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Properties

Label 2-600-8.5-c1-0-28
Degree $2$
Conductor $600$
Sign $-0.951 + 0.306i$
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  + (−1.29 − 0.576i)2-s + i·3-s + (1.33 + 1.48i)4-s + (0.576 − 1.29i)6-s − 1.97·7-s + (−0.867 − 2.69i)8-s − 9-s − 1.43i·11-s + (−1.48 + 1.33i)12-s + 0.241i·13-s + (2.55 + 1.13i)14-s + (−0.430 + 3.97i)16-s − 7.38·17-s + (1.29 + 0.576i)18-s + 3.04i·19-s + ⋯
L(s)  = 1  + (−0.913 − 0.407i)2-s + 0.577i·3-s + (0.667 + 0.744i)4-s + (0.235 − 0.527i)6-s − 0.747·7-s + (−0.306 − 0.951i)8-s − 0.333·9-s − 0.431i·11-s + (−0.429 + 0.385i)12-s + 0.0669i·13-s + (0.682 + 0.304i)14-s + (−0.107 + 0.994i)16-s − 1.79·17-s + (0.304 + 0.135i)18-s + 0.697i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.951 + 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.951 + 0.306i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.951 + 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0121062 - 0.0770205i\)
\(L(\frac12)\) \(\approx\) \(0.0121062 - 0.0770205i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.29 + 0.576i)T \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + 1.97T + 7T^{2} \)
11 \( 1 + 1.43iT - 11T^{2} \)
13 \( 1 - 0.241iT - 13T^{2} \)
17 \( 1 + 7.38T + 17T^{2} \)
19 \( 1 - 3.04iT - 19T^{2} \)
23 \( 1 - 0.874T + 23T^{2} \)
29 \( 1 + 9.07iT - 29T^{2} \)
31 \( 1 + 7.44T + 31T^{2} \)
37 \( 1 + 8.81iT - 37T^{2} \)
41 \( 1 + 1.91T + 41T^{2} \)
43 \( 1 - 11.2iT - 43T^{2} \)
47 \( 1 + 3.34T + 47T^{2} \)
53 \( 1 + 9.20iT - 53T^{2} \)
59 \( 1 + 6.43iT - 59T^{2} \)
61 \( 1 - 4.57iT - 61T^{2} \)
67 \( 1 + 4.86iT - 67T^{2} \)
71 \( 1 + 8.21T + 71T^{2} \)
73 \( 1 + 4.12T + 73T^{2} \)
79 \( 1 + 13.6T + 79T^{2} \)
83 \( 1 - 12.3iT - 83T^{2} \)
89 \( 1 + 8.08T + 89T^{2} \)
97 \( 1 - 10.6T + 97T^{2} \)
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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

−10.14456683275622930985132933489, −9.445174421419753184238895260183, −8.787372763548182735455944764301, −7.85715070620301483133914630459, −6.73181933913039821458155112934, −5.92995485111555063962144940742, −4.30894153375761061127440932509, −3.36253091534764727896667271798, −2.15574677586060905521873215106, −0.05415233351841426558299190794, 1.75515852725344831560134432009, 2.99184795237927914846092328573, 4.78688041449857540650229107298, 5.97272239314117215263523068164, 6.94318582584707543942322173699, 7.22132671330389228548223318515, 8.702467852357201263381642009377, 8.986659782386552030020678846215, 10.11045566183088415851822491329, 10.91174457927097321707303543241

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