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Properties

Label 2-600-120.59-c1-0-50
Degree $2$
Conductor $600$
Sign $-0.527 + 0.849i$
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  − 1.41·2-s + (−0.158 + 1.72i)3-s + 2.00·4-s + (0.224 − 2.43i)6-s − 2.82·8-s + (−2.94 − 0.548i)9-s − 3.78i·11-s + (−0.317 + 3.44i)12-s + 4.00·16-s − 8.02·17-s + (4.17 + 0.775i)18-s − 6.34·19-s + 5.34i·22-s + (0.449 − 4.87i)24-s + (1.41 − 4.99i)27-s + ⋯
L(s)  = 1  − 1.00·2-s + (−0.0917 + 0.995i)3-s + 1.00·4-s + (0.0917 − 0.995i)6-s − 1.00·8-s + (−0.983 − 0.182i)9-s − 1.14i·11-s + (−0.0917 + 0.995i)12-s + 1.00·16-s − 1.94·17-s + (0.983 + 0.182i)18-s − 1.45·19-s + 1.14i·22-s + (0.0917 − 0.995i)24-s + (0.272 − 0.962i)27-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.527 + 0.849i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ -0.527 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0626621 - 0.112649i\)
\(L(\frac12)\) \(\approx\) \(0.0626621 - 0.112649i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41T \)
3 \( 1 + (0.158 - 1.72i)T \)
5 \( 1 \)
good7 \( 1 + 7T^{2} \)
11 \( 1 + 3.78iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 + 8.02T + 17T^{2} \)
19 \( 1 + 6.34T + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + 14.1iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 0.348iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 15.6iT - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 17.0T + 83T^{2} \)
89 \( 1 - 18.4iT - 89T^{2} \)
97 \( 1 - 10iT - 97T^{2} \)
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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

−10.41670138438024737753513993496, −9.413402080148884617202143986953, −8.694513379201643017889231670840, −8.196149719819554742039198400008, −6.65530603530427155152855379083, −6.06255408423310718097842473570, −4.72583909243822732444860140078, −3.51173267978459936367321768806, −2.28980101745073913005824646575, −0.091392521812361061427253712997, 1.77178395313481482222082709951, 2.55974332666153819736857269023, 4.43068662218473563290185324129, 5.96153889278589183998733571304, 6.79214337252876949716857654502, 7.32514714363639507204022811662, 8.446176172849905530420777327093, 8.951077150044081440941298244130, 10.08705707634517204463396311927, 10.98635733053477978392522886992

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