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Properties

Label 2-600-600.29-c0-0-1
Degree $2$
Conductor $600$
Sign $0.929 + 0.368i$
Analytic cond. $0.299439$
Root an. cond. $0.547210$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.951 − 0.309i)5-s + (0.809 − 0.587i)6-s − 1.17i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.587 + 1.80i)11-s + (0.951 − 0.309i)12-s + (0.363 − 1.11i)14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s − i·18-s + ⋯
L(s)  = 1  + (0.951 + 0.309i)2-s + (0.587 − 0.809i)3-s + (0.809 + 0.587i)4-s + (−0.951 − 0.309i)5-s + (0.809 − 0.587i)6-s − 1.17i·7-s + (0.587 + 0.809i)8-s + (−0.309 − 0.951i)9-s + (−0.809 − 0.587i)10-s + (−0.587 + 1.80i)11-s + (0.951 − 0.309i)12-s + (0.363 − 1.11i)14-s + (−0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s − i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.929 + 0.368i$
Analytic conductor: \(0.299439\)
Root analytic conductor: \(0.547210\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (29, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :0),\ 0.929 + 0.368i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.581001672\)
\(L(\frac12)\) \(\approx\) \(1.581001672\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
3 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 + (0.951 + 0.309i)T \)
good7 \( 1 + 1.17iT - T^{2} \)
11 \( 1 + (0.587 - 1.80i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (0.309 - 0.951i)T^{2} \)
19 \( 1 + (-0.309 + 0.951i)T^{2} \)
23 \( 1 + (-0.809 - 0.587i)T^{2} \)
29 \( 1 + (0.309 + 0.951i)T^{2} \)
31 \( 1 + (1.30 - 0.951i)T + (0.309 - 0.951i)T^{2} \)
37 \( 1 + (-0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (0.309 + 0.951i)T^{2} \)
53 \( 1 + (-0.363 + 0.5i)T + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.587 + 1.80i)T + (-0.809 + 0.587i)T^{2} \)
61 \( 1 + (0.809 + 0.587i)T^{2} \)
67 \( 1 + (0.309 - 0.951i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
79 \( 1 + (0.5 + 0.363i)T + (0.309 + 0.951i)T^{2} \)
83 \( 1 + (0.951 + 1.30i)T + (-0.309 + 0.951i)T^{2} \)
89 \( 1 + (0.809 + 0.587i)T^{2} \)
97 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
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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

−11.08318840088090107780214364253, −10.07425996752919593607193674229, −8.738320322643751399583016296997, −7.58389213235734221609752691853, −7.44266877972036742124913547966, −6.64420071256476009707830287119, −5.06327270827912649708680498399, −4.19691112810515798177461400369, −3.30330636615699069099064835231, −1.83921522235390678757505326512, 2.54325087977839769821938099747, 3.24759096176251271887635550111, 4.13910739073987774678598460498, 5.35697832941271490060252648184, 5.96993959085859948239202739192, 7.46816133755269802001550143596, 8.380316672656604286168766163721, 9.128723138257246690176954062734, 10.37401849557678662909036382178, 11.11900594879586665299460782340

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