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Properties

Label 2-600-15.8-c1-0-15
Degree $2$
Conductor $600$
Sign $0.374 + 0.927i$
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  + (1.41 − i)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 + (−1.41 − 5.00i)27-s + 3.65·29-s + 5.65·31-s + (0.828 + 1.17i)33-s + (5.82 − 5.82i)37-s + (−9.24 − 1.58i)39-s + ⋯
L(s)  = 1  + (0.816 − 0.577i)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 + (−0.272 − 0.962i)27-s + 0.679·29-s + 1.01·31-s + (0.144 + 0.203i)33-s + (0.958 − 0.958i)37-s + (−1.48 − 0.253i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(600\)    =    \(2^{3} \cdot 3 \cdot 5^{2}\)
Sign: $0.374 + 0.927i$
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: \((2,\ 600,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67463 - 1.12993i\)
\(L(\frac12)\) \(\approx\) \(1.67463 - 1.12993i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + i)T \)
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

−10.22283882723054447935926358887, −9.814720586610280712371331969874, −8.451297699622461560980585947238, −7.76959572118689690344825836863, −7.33315272223107519543640125363, −6.04924590874431517897680852528, −4.77813925743242695093840840165, −3.73489033304457037078005167146, −2.47561981517998861261580546024, −1.14268760877033019402421744424, 2.00482970560906099925493186154, 2.88826620390397217961820009860, 4.44054691171045969404927856330, 4.95320076997851669360725227616, 6.28601350570573820241886932598, 7.57935936436658637690187808977, 8.296874863021682345630009330297, 9.074247404426216744450170076544, 9.805222157560600812649577247694, 10.70687091568424098218455041385

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