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Properties

Label 2-600-120.53-c1-0-17
Degree $2$
Conductor $600$
Sign $0.209 - 0.977i$
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  + (−0.533 + 1.30i)2-s + (−1.59 + 0.667i)3-s + (−1.43 − 1.39i)4-s + (−0.0218 − 2.44i)6-s + (−0.582 + 0.582i)7-s + (2.59 − 1.13i)8-s + (2.10 − 2.13i)9-s − 3.68·11-s + (3.22 + 1.27i)12-s + (3.88 − 3.88i)13-s + (−0.452 − 1.07i)14-s + (0.0980 + 3.99i)16-s + (0.880 + 0.880i)17-s + (1.66 + 3.90i)18-s + 6.32·19-s + ⋯
L(s)  = 1  + (−0.377 + 0.926i)2-s + (−0.922 + 0.385i)3-s + (−0.715 − 0.698i)4-s + (−0.00892 − 0.999i)6-s + (−0.220 + 0.220i)7-s + (0.916 − 0.399i)8-s + (0.703 − 0.711i)9-s − 1.11·11-s + (0.929 + 0.368i)12-s + (1.07 − 1.07i)13-s + (−0.120 − 0.287i)14-s + (0.0245 + 0.999i)16-s + (0.213 + 0.213i)17-s + (0.393 + 0.919i)18-s + 1.45·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.209 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.209 - 0.977i$
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: \((2,\ 600,\ (\ :1/2),\ 0.209 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.605664 + 0.489857i\)
\(L(\frac12)\) \(\approx\) \(0.605664 + 0.489857i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.533 - 1.30i)T \)
3 \( 1 + (1.59 - 0.667i)T \)
5 \( 1 \)
good7 \( 1 + (0.582 - 0.582i)T - 7iT^{2} \)
11 \( 1 + 3.68T + 11T^{2} \)
13 \( 1 + (-3.88 + 3.88i)T - 13iT^{2} \)
17 \( 1 + (-0.880 - 0.880i)T + 17iT^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 + (2.06 - 2.06i)T - 23iT^{2} \)
29 \( 1 - 1.37iT - 29T^{2} \)
31 \( 1 - 3.32T + 31T^{2} \)
37 \( 1 + (-2.44 - 2.44i)T + 37iT^{2} \)
41 \( 1 - 0.648iT - 41T^{2} \)
43 \( 1 + (-0.819 + 0.819i)T - 43iT^{2} \)
47 \( 1 + (-6.28 - 6.28i)T + 47iT^{2} \)
53 \( 1 + (-5.60 - 5.60i)T + 53iT^{2} \)
59 \( 1 + 6.12iT - 59T^{2} \)
61 \( 1 + 5.13iT - 61T^{2} \)
67 \( 1 + (-4.90 - 4.90i)T + 67iT^{2} \)
71 \( 1 - 4.13iT - 71T^{2} \)
73 \( 1 + (4.69 + 4.69i)T + 73iT^{2} \)
79 \( 1 + 1.10iT - 79T^{2} \)
83 \( 1 + (6.27 + 6.27i)T + 83iT^{2} \)
89 \( 1 - 15.3T + 89T^{2} \)
97 \( 1 + (-5.42 + 5.42i)T - 97iT^{2} \)
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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

−10.56422351901877446547715964337, −10.05400693514599270903798177709, −9.142138813348724985432473412358, −8.041009350572291894827008568960, −7.34801992694191647271205334865, −6.03984822132334667019589722554, −5.66335384379530787911231604332, −4.72704726499784913712228544935, −3.36167219730198779688626683379, −0.928191207604895134203347176830, 0.830433565313843686757394551656, 2.24535658571870897610631819645, 3.70131893040043315990716211383, 4.81298595208822557782034672771, 5.80934179465038007641053089217, 7.04542999125924835927359447702, 7.83527745932852291143831226923, 8.855699335994102901326431165672, 9.970042621877952463649325547468, 10.48242037082996963129249282595

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