Properties

Label 2-600-15.2-c1-0-8
Degree $2$
Conductor $600$
Sign $0.920 - 0.391i$
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  + (1.70 − 0.292i)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 + (4.53 − 2.53i)27-s + 5.65·29-s + 8·31-s + (1.65 + 9.65i)33-s + (−8 − 8i)37-s + 5.65i·41-s + (2 − 2i)43-s + ⋯
L(s)  = 1  + (0.985 − 0.169i)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 + (0.872 − 0.487i)27-s + 1.05·29-s + 1.43·31-s + (0.288 + 1.68i)33-s + (−1.31 − 1.31i)37-s + 0.883i·41-s + (0.304 − 0.304i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.920 - 0.391i$
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: \((2,\ 600,\ (\ :1/2),\ 0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.14924 + 0.437623i\)
\(L(\frac12)\) \(\approx\) \(2.14924 + 0.437623i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.292i)T \)
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

−10.49708862711964756298045257826, −9.788281705732756957387823429899, −8.777494271323086892898375163882, −8.277128103438994246555553898084, −7.25048871863829031643520472450, −6.43561408897137392191719354693, −4.87254526941122579339200741349, −4.22177179584801935656379819514, −2.57669095875863565541452278991, −1.85214976429759715755345088730, 1.31086555829133166954700244857, 2.86690011280739985368394662212, 3.85038367217096879344964794393, 4.81949693080983703297922330155, 6.12257249549774326151727324542, 7.27331507502432617979734143967, 8.212369006360637087168124125263, 8.572840795598017921892430684499, 9.769295788967416636797011897455, 10.52245619461867597643163917101

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