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Properties

Label 2-1500-25.16-c1-0-7
Degree $2$
Conductor $1500$
Sign $0.840 + 0.541i$
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 − 4.62·7-s + (−0.809 + 0.587i)9-s + (4.00 + 2.90i)11-s + (−3.04 + 2.21i)13-s + (0.831 − 2.55i)17-s + (1.81 − 5.58i)19-s + (1.42 + 4.40i)21-s + (5.40 + 3.92i)23-s + (0.809 + 0.587i)27-s + (0.370 + 1.14i)29-s + (1.02 − 3.14i)31-s + (1.52 − 4.70i)33-s + (1.51 − 1.10i)37-s + (3.04 + 2.21i)39-s + ⋯
L(s)  = 1  + (−0.178 − 0.549i)3-s − 1.74·7-s + (−0.269 + 0.195i)9-s + (1.20 + 0.877i)11-s + (−0.844 + 0.613i)13-s + (0.201 − 0.620i)17-s + (0.416 − 1.28i)19-s + (0.311 + 0.960i)21-s + (1.12 + 0.818i)23-s + (0.155 + 0.113i)27-s + (0.0688 + 0.212i)29-s + (0.183 − 0.564i)31-s + (0.266 − 0.819i)33-s + (0.249 − 0.181i)37-s + (0.487 + 0.354i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.840 + 0.541i$
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: \((2,\ 1500,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.229527690\)
\(L(\frac12)\) \(\approx\) \(1.229527690\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.62T + 7T^{2} \)
11 \( 1 + (-4.00 - 2.90i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (3.04 - 2.21i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.831 + 2.55i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.81 + 5.58i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (-5.40 - 3.92i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.370 - 1.14i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-1.02 + 3.14i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.51 + 1.10i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-2.45 + 1.78i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + (0.0801 + 0.246i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.02 + 9.31i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.78 + 5.65i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (5.07 + 3.68i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (0.791 - 2.43i)T + (-54.2 - 39.3i)T^{2} \)
71 \( 1 + (-2.68 - 8.25i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-3.94 - 2.86i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-3.85 - 11.8i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.74 + 8.45i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (11.7 + 8.56i)T + (27.5 + 84.6i)T^{2} \)
97 \( 1 + (-1.23 - 3.79i)T + (-78.4 + 57.0i)T^{2} \)
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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

−9.563089627542844025109028101829, −8.906684357437958949833354352403, −7.32600557276260193099714418841, −7.03847883482216039848372891621, −6.42417040424987354717290068673, −5.35473216516512638048243001815, −4.31805222590964745121834941529, −3.22657149663953879316934832471, −2.30282388207815051312912440397, −0.73372010298634247073675830371, 0.846995443365727127957612879158, 2.84705851182192932030605562997, 3.47467891175904386767544990416, 4.33938474567083418513240207042, 5.67988627797868836850013999054, 6.16612819346988512599634409203, 6.94982716300215758775493841844, 8.026673773000958867346514152023, 9.097613153263694324064647796878, 9.451572643989507006889571219097

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