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Properties

Label 2-600-120.53-c1-0-33
Degree $2$
Conductor $600$
Sign $0.993 - 0.110i$
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 − 0.0941i)2-s + (1.68 + 0.416i)3-s + (1.98 + 0.265i)4-s + (−2.33 − 0.746i)6-s + (0.361 − 0.361i)7-s + (−2.77 − 0.561i)8-s + (2.65 + 1.40i)9-s − 2.63·11-s + (3.22 + 1.27i)12-s + (3.49 − 3.49i)13-s + (−0.544 + 0.476i)14-s + (3.85 + 1.05i)16-s + (3.61 + 3.61i)17-s + (−3.61 − 2.22i)18-s + 0.672·19-s + ⋯
L(s)  = 1  + (−0.997 − 0.0665i)2-s + (0.970 + 0.240i)3-s + (0.991 + 0.132i)4-s + (−0.952 − 0.304i)6-s + (0.136 − 0.136i)7-s + (−0.980 − 0.198i)8-s + (0.884 + 0.466i)9-s − 0.794·11-s + (0.930 + 0.367i)12-s + (0.968 − 0.968i)13-s + (−0.145 + 0.127i)14-s + (0.964 + 0.263i)16-s + (0.876 + 0.876i)17-s + (−0.851 − 0.524i)18-s + 0.154·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.993 - 0.110i$
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: \((2,\ 600,\ (\ :1/2),\ 0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.42061 + 0.0787758i\)
\(L(\frac12)\) \(\approx\) \(1.42061 + 0.0787758i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 + 0.0941i)T \)
3 \( 1 + (-1.68 - 0.416i)T \)
5 \( 1 \)
good7 \( 1 + (-0.361 + 0.361i)T - 7iT^{2} \)
11 \( 1 + 2.63T + 11T^{2} \)
13 \( 1 + (-3.49 + 3.49i)T - 13iT^{2} \)
17 \( 1 + (-3.61 - 3.61i)T + 17iT^{2} \)
19 \( 1 - 0.672T + 19T^{2} \)
23 \( 1 + (-4.31 + 4.31i)T - 23iT^{2} \)
29 \( 1 - 4.76iT - 29T^{2} \)
31 \( 1 - 3.73T + 31T^{2} \)
37 \( 1 + (2.82 + 2.82i)T + 37iT^{2} \)
41 \( 1 + 4.10iT - 41T^{2} \)
43 \( 1 + (7.57 - 7.57i)T - 43iT^{2} \)
47 \( 1 + (-0.987 - 0.987i)T + 47iT^{2} \)
53 \( 1 + (-0.646 - 0.646i)T + 53iT^{2} \)
59 \( 1 + 4.92iT - 59T^{2} \)
61 \( 1 + 6.07iT - 61T^{2} \)
67 \( 1 + (0.349 + 0.349i)T + 67iT^{2} \)
71 \( 1 + 8.63iT - 71T^{2} \)
73 \( 1 + (-11.3 - 11.3i)T + 73iT^{2} \)
79 \( 1 - 4.07iT - 79T^{2} \)
83 \( 1 + (8.53 + 8.53i)T + 83iT^{2} \)
89 \( 1 - 6.58T + 89T^{2} \)
97 \( 1 + (-0.660 + 0.660i)T - 97iT^{2} \)
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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

−10.53633245897462627745899312277, −9.830292482524866035717294741819, −8.760396333619691875683021125210, −8.199581842827251936163753125546, −7.57156926680884230812285766152, −6.43255774583906904131562379607, −5.18593353371116194930067391239, −3.57768850112161648864950391889, −2.76868455813392506192189378834, −1.30496208345299239561107154854, 1.29448400792537772707217103357, 2.54823178778749572254584519464, 3.58333746212578970381938811644, 5.24388026268207287662242193401, 6.54303380975747144707613570309, 7.34619440782252504666114363194, 8.134963657805176055581100283836, 8.844362306720506324449438863512, 9.624444906922216760209048186542, 10.33271873120571083040123758700

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