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Properties

Label 2-1500-15.2-c1-0-5
Degree $2$
Conductor $1500$
Sign $-0.263 - 0.964i$
Analytic cond. $11.9775$
Root an. cond. $3.46086$
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  + (−1.50 − 0.858i)3-s + (3.42 + 3.42i)7-s + (1.52 + 2.58i)9-s + 4.41i·11-s + (2.37 − 2.37i)13-s + (−2.76 + 2.76i)17-s + 5.73i·19-s + (−2.21 − 8.08i)21-s + (−4.22 − 4.22i)23-s + (−0.0829 − 5.19i)27-s − 9.67·29-s − 0.881·31-s + (3.78 − 6.64i)33-s + (−4.89 − 4.89i)37-s + (−5.61 + 1.53i)39-s + ⋯
L(s)  = 1  + (−0.868 − 0.495i)3-s + (1.29 + 1.29i)7-s + (0.509 + 0.860i)9-s + 1.33i·11-s + (0.659 − 0.659i)13-s + (−0.669 + 0.669i)17-s + 1.31i·19-s + (−0.483 − 1.76i)21-s + (−0.880 − 0.880i)23-s + (−0.0159 − 0.999i)27-s − 1.79·29-s − 0.158·31-s + (0.659 − 1.15i)33-s + (−0.804 − 0.804i)37-s + (−0.899 + 0.246i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.263 - 0.964i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.263 - 0.964i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.050818789\)
\(L(\frac12)\) \(\approx\) \(1.050818789\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.50 + 0.858i)T \)
5 \( 1 \)
good7 \( 1 + (-3.42 - 3.42i)T + 7iT^{2} \)
11 \( 1 - 4.41iT - 11T^{2} \)
13 \( 1 + (-2.37 + 2.37i)T - 13iT^{2} \)
17 \( 1 + (2.76 - 2.76i)T - 17iT^{2} \)
19 \( 1 - 5.73iT - 19T^{2} \)
23 \( 1 + (4.22 + 4.22i)T + 23iT^{2} \)
29 \( 1 + 9.67T + 29T^{2} \)
31 \( 1 + 0.881T + 31T^{2} \)
37 \( 1 + (4.89 + 4.89i)T + 37iT^{2} \)
41 \( 1 - 4.74iT - 41T^{2} \)
43 \( 1 + (2.35 - 2.35i)T - 43iT^{2} \)
47 \( 1 + (-1.71 + 1.71i)T - 47iT^{2} \)
53 \( 1 + (-0.0871 - 0.0871i)T + 53iT^{2} \)
59 \( 1 - 12.4T + 59T^{2} \)
61 \( 1 + 8.20T + 61T^{2} \)
67 \( 1 + (5.36 + 5.36i)T + 67iT^{2} \)
71 \( 1 + 5.14iT - 71T^{2} \)
73 \( 1 + (0.0238 - 0.0238i)T - 73iT^{2} \)
79 \( 1 - 10.1iT - 79T^{2} \)
83 \( 1 + (-3.11 - 3.11i)T + 83iT^{2} \)
89 \( 1 + 2.72T + 89T^{2} \)
97 \( 1 + (-4.70 - 4.70i)T + 97iT^{2} \)
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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

−9.836591065966195647982898708803, −8.711341388020179120485135019195, −8.053648289339603103289862274269, −7.41288903026714674326928969156, −6.22976479660565033863444787391, −5.67145764191199251549451402482, −4.91611522386945422254530886440, −3.99645056375004204161999027452, −2.11033066722058758303969974040, −1.71039640348925115926801953957, 0.46804499094674729546263444614, 1.65449909364320620756549892002, 3.53576716925582591614097646440, 4.21344195180676612048642924771, 5.04539501146952301620205935036, 5.82854715108380042181974746107, 6.87793018282578151442114454183, 7.45965158717182287680218711318, 8.583171589391227776006808527222, 9.210919559086762762319344101520

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