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Properties

Label 2-1500-25.16-c1-0-1
Degree $2$
Conductor $1500$
Sign $-0.127 - 0.991i$
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  + (−0.309 − 0.951i)3-s − 1.04·7-s + (−0.809 + 0.587i)9-s + (−5.08 − 3.69i)11-s + (−0.814 + 0.591i)13-s + (1.44 − 4.46i)17-s + (−1.84 + 5.68i)19-s + (0.323 + 0.995i)21-s + (6.51 + 4.73i)23-s + (0.809 + 0.587i)27-s + (2.13 + 6.57i)29-s + (−2.94 + 9.05i)31-s + (−1.94 + 5.98i)33-s + (6.22 − 4.52i)37-s + (0.814 + 0.591i)39-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s − 0.395·7-s + (−0.269 + 0.195i)9-s + (−1.53 − 1.11i)11-s + (−0.225 + 0.164i)13-s + (0.351 − 1.08i)17-s + (−0.423 + 1.30i)19-s + (0.0705 + 0.217i)21-s + (1.35 + 0.986i)23-s + (0.155 + 0.113i)27-s + (0.396 + 1.22i)29-s + (−0.528 + 1.62i)31-s + (−0.338 + 1.04i)33-s + (1.02 − 0.743i)37-s + (0.130 + 0.0947i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.127 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.127 - 0.991i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (1201, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.127 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5135231826\)
\(L(\frac12)\) \(\approx\) \(0.5135231826\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.309 + 0.951i)T \)
5 \( 1 \)
good7 \( 1 + 1.04T + 7T^{2} \)
11 \( 1 + (5.08 + 3.69i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (0.814 - 0.591i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-1.44 + 4.46i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.84 - 5.68i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-6.51 - 4.73i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.13 - 6.57i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (2.94 - 9.05i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.22 + 4.52i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.26 - 0.922i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 9.94T + 43T^{2} \)
47 \( 1 + (-1.49 - 4.60i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (0.873 + 2.68i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.30 - 2.39i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (2.55 + 1.85i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.979 - 3.01i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.01 - 6.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.297 + 0.216i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (1.02 + 3.15i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (4.28 - 13.1i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (-2.02 - 1.47i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (0.0564 + 0.173i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.705734845668037695539795737969, −8.794826194354708453299453635686, −8.024333554822950130224401283302, −7.31569446589206838995702742475, −6.51310051817258927870605299314, −5.46036312016562972629450951646, −5.06636976525899502357577677007, −3.38093181229613093717506923334, −2.80879717874303441504816434732, −1.29706686248960455418939215240, 0.21150122341608556523406446626, 2.24894161057611633592906198355, 3.07337395459705545199567700400, 4.43259967391412840965397151616, 4.90145255029850294467405478005, 5.94376925388069053330004953767, 6.80127886163987849373390431075, 7.73226128282054715064178337035, 8.445161599787492464908863200754, 9.482613901416908113006041156244

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