Properties

Label 2-1500-25.16-c1-0-12
Degree $2$
Conductor $1500$
Sign $0.440 + 0.897i$
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 − 0.595·7-s + (−0.809 + 0.587i)9-s + (−2.71 − 1.97i)11-s + (−3.85 + 2.80i)13-s + (2.31 − 7.11i)17-s + (1.91 − 5.88i)19-s + (−0.184 − 0.566i)21-s + (3.56 + 2.59i)23-s + (−0.809 − 0.587i)27-s + (−0.853 − 2.62i)29-s + (1.38 − 4.26i)31-s + (1.03 − 3.19i)33-s + (1.05 − 0.764i)37-s + (−3.85 − 2.80i)39-s + ⋯
L(s)  = 1  + (0.178 + 0.549i)3-s − 0.225·7-s + (−0.269 + 0.195i)9-s + (−0.818 − 0.594i)11-s + (−1.07 + 0.777i)13-s + (0.560 − 1.72i)17-s + (0.438 − 1.35i)19-s + (−0.0401 − 0.123i)21-s + (0.743 + 0.540i)23-s + (−0.155 − 0.113i)27-s + (−0.158 − 0.487i)29-s + (0.249 − 0.766i)31-s + (0.180 − 0.555i)33-s + (0.172 − 0.125i)37-s + (−0.617 − 0.448i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.440 + 0.897i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.440 + 0.897i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.440 + 0.897i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.192181682\)
\(L(\frac12)\) \(\approx\) \(1.192181682\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.595T + 7T^{2} \)
11 \( 1 + (2.71 + 1.97i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.85 - 2.80i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-2.31 + 7.11i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.91 + 5.88i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-3.56 - 2.59i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (0.853 + 2.62i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.38 + 4.26i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.05 + 0.764i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-7.61 + 5.53i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 7.59T + 43T^{2} \)
47 \( 1 + (1.36 + 4.18i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-2.53 - 7.80i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-1.80 + 1.31i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (10.2 + 7.41i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (2.58 - 7.94i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.09 - 6.46i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (5.91 + 4.29i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (3.26 + 10.0i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (1.29 - 3.97i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (3.43 + 2.49i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (3.79 + 11.6i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.454480849220681074936602674753, −8.784255754241373673141584411030, −7.46954897561464331502156693931, −7.27135079153999451676898112719, −5.88386138328686280922623734274, −5.07405293599708704516294632820, −4.41564616270635524756139432988, −3.05167336156184873347666265567, −2.51551378928971543711551379264, −0.47396694980402000315482771990, 1.34019337291717782808782549461, 2.54930982178034796428894937137, 3.43254035236809412117262907470, 4.67257977706005213053429805465, 5.60946155094616276310057716265, 6.33967886540388132669959985922, 7.49302703425323516896389244221, 7.82301866339609199358016439218, 8.664919462163359467822269451942, 9.792868512607957869477840387742

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