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Properties

Label 2-600-15.2-c1-0-3
Degree $2$
Conductor $600$
Sign $0.662 - 0.749i$
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 − 1.41i)3-s + (2.41 + 2.41i)7-s + (−1.00 + 2.82i)9-s + 0.828i·11-s + (−3.82 + 3.82i)13-s + (−1.82 + 1.82i)17-s − 0.828i·19-s + (1 − 5.82i)21-s + (4.41 + 4.41i)23-s + (5.00 − 1.41i)27-s − 3.65·29-s + 5.65·31-s + (1.17 − 0.828i)33-s + (5.82 + 5.82i)37-s + (9.24 + 1.58i)39-s + ⋯
L(s)  = 1  + (−0.577 − 0.816i)3-s + (0.912 + 0.912i)7-s + (−0.333 + 0.942i)9-s + 0.249i·11-s + (−1.06 + 1.06i)13-s + (−0.443 + 0.443i)17-s − 0.190i·19-s + (0.218 − 1.27i)21-s + (0.920 + 0.920i)23-s + (0.962 − 0.272i)27-s − 0.679·29-s + 1.01·31-s + (0.203 − 0.144i)33-s + (0.958 + 0.958i)37-s + (1.48 + 0.253i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.662 - 0.749i$
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: \((2,\ 600,\ (\ :1/2),\ 0.662 - 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00208 + 0.451901i\)
\(L(\frac12)\) \(\approx\) \(1.00208 + 0.451901i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 1.41i)T \)
5 \( 1 \)
good7 \( 1 + (-2.41 - 2.41i)T + 7iT^{2} \)
11 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (3.82 - 3.82i)T - 13iT^{2} \)
17 \( 1 + (1.82 - 1.82i)T - 17iT^{2} \)
19 \( 1 + 0.828iT - 19T^{2} \)
23 \( 1 + (-4.41 - 4.41i)T + 23iT^{2} \)
29 \( 1 + 3.65T + 29T^{2} \)
31 \( 1 - 5.65T + 31T^{2} \)
37 \( 1 + (-5.82 - 5.82i)T + 37iT^{2} \)
41 \( 1 - 5.65iT - 41T^{2} \)
43 \( 1 + (-0.414 + 0.414i)T - 43iT^{2} \)
47 \( 1 + (-3.58 + 3.58i)T - 47iT^{2} \)
53 \( 1 + (3 + 3i)T + 53iT^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 - 0.343T + 61T^{2} \)
67 \( 1 + (10.0 + 10.0i)T + 67iT^{2} \)
71 \( 1 - 10.4iT - 71T^{2} \)
73 \( 1 + (-4.65 + 4.65i)T - 73iT^{2} \)
79 \( 1 - 0.828iT - 79T^{2} \)
83 \( 1 + (-3.24 - 3.24i)T + 83iT^{2} \)
89 \( 1 + 15.6T + 89T^{2} \)
97 \( 1 + (1 + i)T + 97iT^{2} \)
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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

−11.21492003146416735206335097261, −9.895761738235992788185038009864, −8.935060833768308025572048299284, −8.036870110476978599291607003729, −7.19178016952799043732192889160, −6.31975856188716703070499755179, −5.24651245417923246231817363517, −4.56503433467273257658051402684, −2.56377605174596560865063889237, −1.60688106022990730191255505391, 0.68495315058353425993022482270, 2.78991191725055461258056631825, 4.17894441372168539200704845626, 4.86474228005612123630928045181, 5.76540175381736131906474234092, 7.02092135956106898309799424082, 7.84791414692095660375457161460, 8.918494841169426759024384434722, 9.879818515762060843798867371100, 10.67586178188134931240081450454

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