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Properties

Label 2-600-120.53-c1-0-29
Degree $2$
Conductor $600$
Sign $-0.816 + 0.577i$
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  + (−0.722 − 1.21i)2-s + (−1.48 − 0.889i)3-s + (−0.956 + 1.75i)4-s + (−0.00827 + 2.44i)6-s + (−2.09 + 2.09i)7-s + (2.82 − 0.106i)8-s + (1.41 + 2.64i)9-s + 2.65·11-s + (2.98 − 1.75i)12-s + (4.21 − 4.21i)13-s + (4.05 + 1.03i)14-s + (−2.17 − 3.35i)16-s + (−3.84 − 3.84i)17-s + (2.19 − 3.63i)18-s − 3.15·19-s + ⋯
L(s)  = 1  + (−0.510 − 0.859i)2-s + (−0.857 − 0.513i)3-s + (−0.478 + 0.878i)4-s + (−0.00337 + 0.999i)6-s + (−0.790 + 0.790i)7-s + (0.999 − 0.0377i)8-s + (0.472 + 0.881i)9-s + 0.800·11-s + (0.861 − 0.507i)12-s + (1.16 − 1.16i)13-s + (1.08 + 0.275i)14-s + (−0.542 − 0.839i)16-s + (−0.933 − 0.933i)17-s + (0.516 − 0.856i)18-s − 0.724·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $-0.816 + 0.577i$
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: \((2,\ 600,\ (\ :1/2),\ -0.816 + 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.172487 - 0.542092i\)
\(L(\frac12)\) \(\approx\) \(0.172487 - 0.542092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.722 + 1.21i)T \)
3 \( 1 + (1.48 + 0.889i)T \)
5 \( 1 \)
good7 \( 1 + (2.09 - 2.09i)T - 7iT^{2} \)
11 \( 1 - 2.65T + 11T^{2} \)
13 \( 1 + (-4.21 + 4.21i)T - 13iT^{2} \)
17 \( 1 + (3.84 + 3.84i)T + 17iT^{2} \)
19 \( 1 + 3.15T + 19T^{2} \)
23 \( 1 + (1.60 - 1.60i)T - 23iT^{2} \)
29 \( 1 + 4.35iT - 29T^{2} \)
31 \( 1 - 1.56T + 31T^{2} \)
37 \( 1 + (-4.94 - 4.94i)T + 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (-0.219 + 0.219i)T - 43iT^{2} \)
47 \( 1 + (-1.83 - 1.83i)T + 47iT^{2} \)
53 \( 1 + (4.64 + 4.64i)T + 53iT^{2} \)
59 \( 1 + 9.93iT - 59T^{2} \)
61 \( 1 + 11.9iT - 61T^{2} \)
67 \( 1 + (8.80 + 8.80i)T + 67iT^{2} \)
71 \( 1 - 2.89iT - 71T^{2} \)
73 \( 1 + (-2.29 - 2.29i)T + 73iT^{2} \)
79 \( 1 + 9.02iT - 79T^{2} \)
83 \( 1 + (9.78 + 9.78i)T + 83iT^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 + (-9.26 + 9.26i)T - 97iT^{2} \)
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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

−10.46540716403162269662354901125, −9.539867081427969308145104491111, −8.716723212836545759946191882768, −7.81179716433197073963458020724, −6.61473679413817790633033237283, −5.90339120063926378857575530143, −4.59318267638006589735496646753, −3.30839841141701178997403847234, −2.04550221452305117514288582270, −0.47708305792601343074337324835, 1.28206261100478268773277279177, 4.01422515049213329485551764974, 4.35966570305007480343921843440, 6.04372567311035992392105122833, 6.40455584128997668038343242091, 7.11774166694443930476485137562, 8.621244841928996328129438895559, 9.187980930025542846392535036339, 10.13249013475091627254804312783, 10.80076907099193955577747361338

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