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Properties

Label 2-600-120.53-c1-0-34
Degree $2$
Conductor $600$
Sign $0.235 + 0.971i$
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  + (−1.37 + 0.313i)2-s + (−0.378 − 1.69i)3-s + (1.80 − 0.865i)4-s + (1.05 + 2.21i)6-s + (−0.699 + 0.699i)7-s + (−2.21 + 1.75i)8-s + (−2.71 + 1.27i)9-s + 4.24·11-s + (−2.14 − 2.71i)12-s + (2.08 − 2.08i)13-s + (0.745 − 1.18i)14-s + (2.50 − 3.12i)16-s + (−0.0541 − 0.0541i)17-s + (3.33 − 2.61i)18-s + 4.52·19-s + ⋯
L(s)  = 1  + (−0.975 + 0.221i)2-s + (−0.218 − 0.975i)3-s + (0.901 − 0.432i)4-s + (0.429 + 0.902i)6-s + (−0.264 + 0.264i)7-s + (−0.783 + 0.621i)8-s + (−0.904 + 0.426i)9-s + 1.27·11-s + (−0.619 − 0.785i)12-s + (0.577 − 0.577i)13-s + (0.199 − 0.316i)14-s + (0.625 − 0.780i)16-s + (−0.0131 − 0.0131i)17-s + (0.787 − 0.616i)18-s + 1.03·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.235 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.235 + 0.971i$
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: \((2,\ 600,\ (\ :1/2),\ 0.235 + 0.971i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.678368 - 0.533806i\)
\(L(\frac12)\) \(\approx\) \(0.678368 - 0.533806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.37 - 0.313i)T \)
3 \( 1 + (0.378 + 1.69i)T \)
5 \( 1 \)
good7 \( 1 + (0.699 - 0.699i)T - 7iT^{2} \)
11 \( 1 - 4.24T + 11T^{2} \)
13 \( 1 + (-2.08 + 2.08i)T - 13iT^{2} \)
17 \( 1 + (0.0541 + 0.0541i)T + 17iT^{2} \)
19 \( 1 - 4.52T + 19T^{2} \)
23 \( 1 + (-3.48 + 3.48i)T - 23iT^{2} \)
29 \( 1 + 3.90iT - 29T^{2} \)
31 \( 1 + 10.2T + 31T^{2} \)
37 \( 1 + (7.29 + 7.29i)T + 37iT^{2} \)
41 \( 1 + 8.74iT - 41T^{2} \)
43 \( 1 + (-4.60 + 4.60i)T - 43iT^{2} \)
47 \( 1 + (-8.05 - 8.05i)T + 47iT^{2} \)
53 \( 1 + (0.260 + 0.260i)T + 53iT^{2} \)
59 \( 1 + 2.89iT - 59T^{2} \)
61 \( 1 + 6.70iT - 61T^{2} \)
67 \( 1 + (-1.75 - 1.75i)T + 67iT^{2} \)
71 \( 1 + 10.5iT - 71T^{2} \)
73 \( 1 + (4.53 + 4.53i)T + 73iT^{2} \)
79 \( 1 - 1.46iT - 79T^{2} \)
83 \( 1 + (-5.61 - 5.61i)T + 83iT^{2} \)
89 \( 1 - 5.42T + 89T^{2} \)
97 \( 1 + (-11.3 + 11.3i)T - 97iT^{2} \)
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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

−10.62410331867661320537293508126, −9.157911820785618986803282110002, −8.938287449098902397625086532950, −7.68219859941346591611108843202, −7.08673722356937083769492330102, −6.14984641990126936809503085598, −5.47136364247767631143278722488, −3.42908402112188443696847347656, −2.05125866145732039282867729988, −0.77773600240133215170587911737, 1.33342271952135815771463004244, 3.23268489955071970376960127341, 3.92824604655008797173256820096, 5.42003364561254249505291900477, 6.52786907680827767228688245565, 7.29502433866554382329357202141, 8.701854487485925015582779807885, 9.165686078161304705959650363664, 9.863446890086587536193135524587, 10.75286079467825322889884421825

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