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Properties

Label 2-600-120.53-c1-0-28
Degree $2$
Conductor $600$
Sign $0.439 + 0.898i$
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  + (−0.552 − 1.30i)2-s + (−0.504 + 1.65i)3-s + (−1.38 + 1.43i)4-s + (2.43 − 0.257i)6-s + (2.83 − 2.83i)7-s + (2.64 + 1.01i)8-s + (−2.48 − 1.67i)9-s − 4.74·11-s + (−1.68 − 3.02i)12-s + (0.867 − 0.867i)13-s + (−5.26 − 2.12i)14-s + (−0.136 − 3.99i)16-s + (1.73 + 1.73i)17-s + (−0.803 + 4.16i)18-s + 3.35·19-s + ⋯
L(s)  = 1  + (−0.390 − 0.920i)2-s + (−0.291 + 0.956i)3-s + (−0.694 + 0.719i)4-s + (0.994 − 0.105i)6-s + (1.07 − 1.07i)7-s + (0.933 + 0.358i)8-s + (−0.829 − 0.557i)9-s − 1.43·11-s + (−0.485 − 0.874i)12-s + (0.240 − 0.240i)13-s + (−1.40 − 0.568i)14-s + (−0.0341 − 0.999i)16-s + (0.419 + 0.419i)17-s + (−0.189 + 0.981i)18-s + 0.769·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.439 + 0.898i$
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: \((2,\ 600,\ (\ :1/2),\ 0.439 + 0.898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865605 - 0.540172i\)
\(L(\frac12)\) \(\approx\) \(0.865605 - 0.540172i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.552 + 1.30i)T \)
3 \( 1 + (0.504 - 1.65i)T \)
5 \( 1 \)
good7 \( 1 + (-2.83 + 2.83i)T - 7iT^{2} \)
11 \( 1 + 4.74T + 11T^{2} \)
13 \( 1 + (-0.867 + 0.867i)T - 13iT^{2} \)
17 \( 1 + (-1.73 - 1.73i)T + 17iT^{2} \)
19 \( 1 - 3.35T + 19T^{2} \)
23 \( 1 + (-4.82 + 4.82i)T - 23iT^{2} \)
29 \( 1 + 0.936iT - 29T^{2} \)
31 \( 1 - 5.49T + 31T^{2} \)
37 \( 1 + (0.749 + 0.749i)T + 37iT^{2} \)
41 \( 1 + 2.24iT - 41T^{2} \)
43 \( 1 + (-8.75 + 8.75i)T - 43iT^{2} \)
47 \( 1 + (6.71 + 6.71i)T + 47iT^{2} \)
53 \( 1 + (-8.46 - 8.46i)T + 53iT^{2} \)
59 \( 1 + 6.53iT - 59T^{2} \)
61 \( 1 - 5.10iT - 61T^{2} \)
67 \( 1 + (-4.85 - 4.85i)T + 67iT^{2} \)
71 \( 1 - 8.34iT - 71T^{2} \)
73 \( 1 + (-1.57 - 1.57i)T + 73iT^{2} \)
79 \( 1 + 7.75iT - 79T^{2} \)
83 \( 1 + (9.15 + 9.15i)T + 83iT^{2} \)
89 \( 1 - 4.97T + 89T^{2} \)
97 \( 1 + (1.42 - 1.42i)T - 97iT^{2} \)
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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

−10.39946337533528825335212288316, −10.18286073211884767628990443720, −8.843023988481793514926274739581, −8.130999955035264504731261797138, −7.28459023530573117422293378858, −5.44709198166272401730697451131, −4.73066789775269279037286744633, −3.82306664661114721535043675491, −2.65819789935031609477530947404, −0.804384037445412693136918861323, 1.28049228262356080064144734601, 2.69226569770743949228499718391, 5.03442836456117923062315796262, 5.32309693752516102691733137790, 6.33337300359978549067662135094, 7.53174940822344159235929412550, 7.927853468239414797970340926578, 8.738227610746490741906612065185, 9.718481716810130149675936239808, 10.96910901027302441562565564274

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