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Properties

Label 2-1500-25.6-c1-0-8
Degree $2$
Conductor $1500$
Sign $0.982 - 0.186i$
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  + (0.809 − 0.587i)3-s + 1.57·7-s + (0.309 − 0.951i)9-s + (1.19 + 3.69i)11-s + (−0.106 + 0.326i)13-s + (4.91 + 3.56i)17-s + (−2.98 − 2.17i)19-s + (1.27 − 0.928i)21-s + (0.429 + 1.32i)23-s + (−0.309 − 0.951i)27-s + (2.69 − 1.95i)29-s + (4.25 + 3.08i)31-s + (3.13 + 2.28i)33-s + (−2.64 + 8.14i)37-s + (0.106 + 0.326i)39-s + ⋯
L(s)  = 1  + (0.467 − 0.339i)3-s + 0.596·7-s + (0.103 − 0.317i)9-s + (0.361 + 1.11i)11-s + (−0.0294 + 0.0905i)13-s + (1.19 + 0.865i)17-s + (−0.685 − 0.497i)19-s + (0.278 − 0.202i)21-s + (0.0895 + 0.275i)23-s + (−0.0594 − 0.183i)27-s + (0.500 − 0.363i)29-s + (0.763 + 0.554i)31-s + (0.546 + 0.397i)33-s + (−0.435 + 1.33i)37-s + (0.0169 + 0.0522i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.982 - 0.186i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (901, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.982 - 0.186i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.234100766\)
\(L(\frac12)\) \(\approx\) \(2.234100766\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.809 + 0.587i)T \)
5 \( 1 \)
good7 \( 1 - 1.57T + 7T^{2} \)
11 \( 1 + (-1.19 - 3.69i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (0.106 - 0.326i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.91 - 3.56i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.98 + 2.17i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.429 - 1.32i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-2.69 + 1.95i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-4.25 - 3.08i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.64 - 8.14i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.394 - 1.21i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 1.42T + 43T^{2} \)
47 \( 1 + (-0.303 + 0.220i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-9.14 + 6.64i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.57 + 11.0i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.38 + 10.4i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-8.46 - 6.14i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (-8.19 + 5.95i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.07 - 12.5i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (11.1 - 8.11i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-3.74 - 2.71i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 + (2.24 + 6.91i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-4.89 + 3.55i)T + (29.9 - 92.2i)T^{2} \)
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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

−9.645930189210058852013429655039, −8.360252932500907980321623212267, −8.181137093090679306051426292017, −7.04761751868272570835560602673, −6.49364690289232294980838685683, −5.26716470088119732550798497079, −4.44271401322371975663245486418, −3.45900092198303976844940327788, −2.23690792839424728798875527602, −1.30321098761400624839358353922, 0.995150024767122146050124145370, 2.44512044854759710597344477128, 3.42600839070876646913812037072, 4.30216187159879689603639317603, 5.32282696239061482702530830292, 6.06090291776931559298063884451, 7.21030168242445438418613126439, 8.002883578420320282168148993976, 8.647846822098613432763532098383, 9.340263397629835933115470696230

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