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Properties

Label 2-1500-15.2-c1-0-11
Degree $2$
Conductor $1500$
Sign $-0.278 - 0.960i$
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  + (−0.835 + 1.51i)3-s + (1.05 + 1.05i)7-s + (−1.60 − 2.53i)9-s + 5.36i·11-s + (2.50 − 2.50i)13-s + (4.79 − 4.79i)17-s + 2.75i·19-s + (−2.48 + 0.719i)21-s + (3.68 + 3.68i)23-s + (5.18 − 0.316i)27-s + 3.14·29-s − 9.41·31-s + (−8.13 − 4.48i)33-s + (3.00 + 3.00i)37-s + (1.71 + 5.90i)39-s + ⋯
L(s)  = 1  + (−0.482 + 0.875i)3-s + (0.399 + 0.399i)7-s + (−0.534 − 0.845i)9-s + 1.61i·11-s + (0.696 − 0.696i)13-s + (1.16 − 1.16i)17-s + 0.631i·19-s + (−0.542 + 0.157i)21-s + (0.769 + 0.769i)23-s + (0.998 − 0.0608i)27-s + 0.583·29-s − 1.69·31-s + (−1.41 − 0.780i)33-s + (0.494 + 0.494i)37-s + (0.274 + 0.945i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.278 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.278 - 0.960i$
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: \((2,\ 1500,\ (\ :1/2),\ -0.278 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.427920862\)
\(L(\frac12)\) \(\approx\) \(1.427920862\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.835 - 1.51i)T \)
5 \( 1 \)
good7 \( 1 + (-1.05 - 1.05i)T + 7iT^{2} \)
11 \( 1 - 5.36iT - 11T^{2} \)
13 \( 1 + (-2.50 + 2.50i)T - 13iT^{2} \)
17 \( 1 + (-4.79 + 4.79i)T - 17iT^{2} \)
19 \( 1 - 2.75iT - 19T^{2} \)
23 \( 1 + (-3.68 - 3.68i)T + 23iT^{2} \)
29 \( 1 - 3.14T + 29T^{2} \)
31 \( 1 + 9.41T + 31T^{2} \)
37 \( 1 + (-3.00 - 3.00i)T + 37iT^{2} \)
41 \( 1 - 10.8iT - 41T^{2} \)
43 \( 1 + (1.30 - 1.30i)T - 43iT^{2} \)
47 \( 1 + (3.22 - 3.22i)T - 47iT^{2} \)
53 \( 1 + (0.347 + 0.347i)T + 53iT^{2} \)
59 \( 1 - 4.13T + 59T^{2} \)
61 \( 1 - 3.72T + 61T^{2} \)
67 \( 1 + (8.11 + 8.11i)T + 67iT^{2} \)
71 \( 1 - 6.87iT - 71T^{2} \)
73 \( 1 + (1.19 - 1.19i)T - 73iT^{2} \)
79 \( 1 + 7.55iT - 79T^{2} \)
83 \( 1 + (-8.29 - 8.29i)T + 83iT^{2} \)
89 \( 1 + 17.1T + 89T^{2} \)
97 \( 1 + (-7.70 - 7.70i)T + 97iT^{2} \)
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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

−9.737332376344437112614517167909, −9.207003522501617518070771610870, −8.107057363530596692585133299034, −7.36748075940472170967507451727, −6.32879537431041873557803927022, −5.28663036456738167438279536735, −4.96527107254567050457570186458, −3.80739543272338324287817512826, −2.88162148721206474508066634961, −1.32846207606500626707081585388, 0.68516314176584618115784690449, 1.70432622879750743799968724235, 3.10575862201813701375658687173, 4.10752091286435969254905129017, 5.40610806496815923387971616097, 5.94476145098647912130730347544, 6.80706595024446820423598499514, 7.58137541998785096822332953023, 8.508100590996742466180522858854, 8.853970336986447983233960966696

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