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Properties

Label 2-1500-25.16-c1-0-4
Degree $2$
Conductor $1500$
Sign $0.604 - 0.796i$
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 − 0.547·7-s + (−0.809 + 0.587i)9-s + (1.08 + 0.786i)11-s + (0.244 − 0.177i)13-s + (−1.24 + 3.84i)17-s + (−1.74 + 5.35i)19-s + (0.169 + 0.520i)21-s + (0.198 + 0.144i)23-s + (0.809 + 0.587i)27-s + (−0.423 − 1.30i)29-s + (−1.09 + 3.36i)31-s + (0.413 − 1.27i)33-s + (−1.76 + 1.28i)37-s + (−0.244 − 0.177i)39-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s − 0.206·7-s + (−0.269 + 0.195i)9-s + (0.326 + 0.237i)11-s + (0.0677 − 0.0492i)13-s + (−0.302 + 0.932i)17-s + (−0.399 + 1.22i)19-s + (0.0369 + 0.113i)21-s + (0.0413 + 0.0300i)23-s + (0.155 + 0.113i)27-s + (−0.0785 − 0.241i)29-s + (−0.196 + 0.604i)31-s + (0.0719 − 0.221i)33-s + (−0.290 + 0.211i)37-s + (−0.0391 − 0.0284i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.604 - 0.796i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.604 - 0.796i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.604 - 0.796i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.221601465\)
\(L(\frac12)\) \(\approx\) \(1.221601465\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.547T + 7T^{2} \)
11 \( 1 + (-1.08 - 0.786i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-0.244 + 0.177i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.24 - 3.84i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (1.74 - 5.35i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-0.198 - 0.144i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.423 + 1.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.09 - 3.36i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (1.76 - 1.28i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.93 + 5.76i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.35T + 43T^{2} \)
47 \( 1 + (-3.23 - 9.96i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.37 - 7.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.35 - 2.44i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.67 + 1.22i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.62 - 8.07i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.83 + 8.73i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-8.86 - 6.43i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-4.69 - 14.4i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.05 + 6.32i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (4.02 + 2.92i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (1.25 + 3.86i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.533182681175478188915115589433, −8.757795578715613172932822744805, −7.923802996065443133794662110320, −7.22559290416683580847551167785, −6.21838955067818002981123564457, −5.79450945496154620081467594341, −4.49313235174428903818823550927, −3.63905847885217444518316842616, −2.35761592232203087168294287782, −1.27379160494936879841490351163, 0.53229606632592934342205841440, 2.31692683949718179942229602436, 3.35028068049908787383578884084, 4.36253877812082378462740187307, 5.10421620447838086047853081294, 6.10104622366088078318232530366, 6.86221318913381944466302377357, 7.74066871528522013170133003496, 8.879276962812296750648103379638, 9.259294609319294397420176997768

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