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Properties

Label 2-1500-25.11-c1-0-9
Degree $2$
Conductor $1500$
Sign $0.860 + 0.509i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.860 + 0.509i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (301, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.860 + 0.509i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.422710153\)
\(L(\frac12)\) \(\approx\) \(2.422710153\)
\(L(\frac{3}{2})\) 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

−9.168779023589679762383619988237, −8.444513370293682014134606572594, −8.006175611614118904505928768799, −7.05844792881612108395769447162, −6.14180068074820905674598997476, −5.39019342379641320295078062568, −4.24339911244471330083291813094, −3.46468660921421265154250346828, −1.84615160184018772871705311362, −1.27143172375043871225346644578, 1.29670319197013289277482486641, 2.32457949699020796982796466780, 3.81153200068901882270209550448, 4.46450511953753359493319776652, 5.19842321304958439315791624819, 6.25782251755041905853713055966, 7.22353276431707022918634214152, 8.222188092139699418675662620186, 8.596257869985036622635815519009, 9.514009579948707364609544256837

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