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Properties

Label 2-3000-24.5-c0-0-12
Degree $2$
Conductor $3000$
Sign $1$
Analytic cond. $1.49719$
Root an. cond. $1.22359$
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  + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s − 0.618·11-s − i·12-s − 1.61i·13-s + 16-s − 1.61i·17-s − i·18-s − 0.618i·22-s − 0.618i·23-s + 24-s + ⋯
L(s)  = 1  + i·2-s + i·3-s − 4-s − 6-s − i·8-s − 9-s − 0.618·11-s − i·12-s − 1.61i·13-s + 16-s − 1.61i·17-s − i·18-s − 0.618i·22-s − 0.618i·23-s + 24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3000\)    =    \(2^{3} \cdot 3 \cdot 5^{3}\)
Sign: $1$
Analytic conductor: \(1.49719\)
Root analytic conductor: \(1.22359\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3000} (2501, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3000,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6337150071\)
\(L(\frac12)\) \(\approx\) \(0.6337150071\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 - iT \)
5 \( 1 \)
good7 \( 1 + T^{2} \)
11 \( 1 + 0.618T + T^{2} \)
13 \( 1 + 1.61iT - T^{2} \)
17 \( 1 + 1.61iT - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 0.618iT - T^{2} \)
29 \( 1 + 1.61T + T^{2} \)
31 \( 1 - 0.618T + T^{2} \)
37 \( 1 + 0.618iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 0.618iT - T^{2} \)
47 \( 1 + 1.61iT - T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + 1.61T + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - 1.61iT - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 - 1.61T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.906609289613079372957572183580, −8.042455131951304710764753121467, −7.60933751161767848633789734327, −6.60250083578198930953071448477, −5.55806267412328746995464962930, −5.28027451744770670558395144279, −4.48001195707926905185271710867, −3.46302573654444977406016926780, −2.72138162976815785165794055604, −0.39133585430441749381621944950, 1.52953448317087691110654397746, 2.01079151461349706238624390780, 3.14725204987242702228728696487, 4.01621930066492584043331091809, 4.95793805511434338215085702895, 5.92396404271262933102473759447, 6.56010354428797255821423360697, 7.70247955250251179754081613891, 8.105397061463694308312303097442, 9.073632414706238269986022242723

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