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Properties

Label 2-600-15.2-c1-0-13
Degree $2$
Conductor $600$
Sign $0.0618 + 0.998i$
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  + (0.292 − 1.70i)3-s + (2 + 2i)7-s + (−2.82 − i)9-s − 5.65i·11-s + (2.82 − 2.82i)17-s − 4i·19-s + (4 − 2.82i)21-s + (4.24 + 4.24i)23-s + (−2.53 + 4.53i)27-s − 5.65·29-s + 8·31-s + (−9.65 − 1.65i)33-s + (−8 − 8i)37-s − 5.65i·41-s + (2 − 2i)43-s + ⋯
L(s)  = 1  + (0.169 − 0.985i)3-s + (0.755 + 0.755i)7-s + (−0.942 − 0.333i)9-s − 1.70i·11-s + (0.685 − 0.685i)17-s − 0.917i·19-s + (0.872 − 0.617i)21-s + (0.884 + 0.884i)23-s + (−0.487 + 0.872i)27-s − 1.05·29-s + 1.43·31-s + (−1.68 − 0.288i)33-s + (−1.31 − 1.31i)37-s − 0.883i·41-s + (0.304 − 0.304i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.0618 + 0.998i$
Analytic conductor: \(4.79102\)
Root analytic conductor: \(2.18884\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{600} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.0618 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.15136 - 1.08221i\)
\(L(\frac12)\) \(\approx\) \(1.15136 - 1.08221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.292 + 1.70i)T \)
5 \( 1 \)
good7 \( 1 + (-2 - 2i)T + 7iT^{2} \)
11 \( 1 + 5.65iT - 11T^{2} \)
13 \( 1 - 13iT^{2} \)
17 \( 1 + (-2.82 + 2.82i)T - 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (-4.24 - 4.24i)T + 23iT^{2} \)
29 \( 1 + 5.65T + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 + (8 + 8i)T + 37iT^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (-2 + 2i)T - 43iT^{2} \)
47 \( 1 + (1.41 - 1.41i)T - 47iT^{2} \)
53 \( 1 + (-5.65 - 5.65i)T + 53iT^{2} \)
59 \( 1 + 5.65T + 59T^{2} \)
61 \( 1 + 6T + 61T^{2} \)
67 \( 1 + (-6 - 6i)T + 67iT^{2} \)
71 \( 1 - 11.3iT - 71T^{2} \)
73 \( 1 + (8 - 8i)T - 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + (-9.89 - 9.89i)T + 83iT^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + (-8 - 8i)T + 97iT^{2} \)
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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

−10.76474368085959758832814518758, −9.148108900018671106456004147899, −8.727355781855999880679092779502, −7.81375719714500017868336676411, −6.99183340137787615935079542034, −5.77765085315488845289309526320, −5.27487929781584120190708097856, −3.42898663251707303857578393591, −2.43534831154197271873436270643, −0.936459502804013404569416563802, 1.75838974038615239535563466949, 3.34470924901840740743919297413, 4.47885609832175514594065076086, 4.93579949982330244563366460651, 6.32615977138206328877332237972, 7.54434022681102129259711095366, 8.198736528586330265477450939756, 9.288907941500785219884959084649, 10.26790997470103556179578276889, 10.45190105737124379120231919022

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