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Properties

Label 2-600-15.8-c1-0-1
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: $\chi_{600} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 600,\ (\ :1/2),\ 0.0618 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.514907 + 0.483979i\)
\(L(\frac12)\) \(\approx\) \(0.514907 + 0.483979i\)
\(L(\frac{3}{2})\) 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

−11.04229972753002009675876511103, −9.753997643076284633844079957989, −9.301672229605101429192155534231, −7.998944319359414530646437006718, −7.33084295128124573132817398062, −6.38350796570949937163100000521, −5.64189303461198336316953931195, −4.37009310926899424655781481218, −2.78047101899898264838600938747, −1.80731057618865983594773881735, 0.38447119718784428726770885984, 2.87984528571417189245894499157, 3.78386079499813411511917271885, 4.69356754640700969671639254612, 6.03378294820230431158942823606, 6.50791660362568492561762371584, 8.057712679896400733452429619945, 8.791832748349670179108095277248, 9.688495470443984327166757642219, 10.52112988028441136435944972993

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