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Properties

Label 2-1500-25.4-c1-0-3
Degree $2$
Conductor $1500$
Sign $0.296 - 0.955i$
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  + (0.587 − 0.809i)3-s + 4.32i·7-s + (−0.309 − 0.951i)9-s + (0.180 − 0.555i)11-s + (0.918 − 0.298i)13-s + (1.36 + 1.88i)17-s + (−4.35 + 3.16i)19-s + (3.49 + 2.54i)21-s + (1.29 + 0.419i)23-s + (−0.951 − 0.309i)27-s + (−0.571 − 0.415i)29-s + (−6.86 + 4.98i)31-s + (−0.343 − 0.472i)33-s + (5.81 − 1.89i)37-s + (0.298 − 0.918i)39-s + ⋯
L(s)  = 1  + (0.339 − 0.467i)3-s + 1.63i·7-s + (−0.103 − 0.317i)9-s + (0.0544 − 0.167i)11-s + (0.254 − 0.0828i)13-s + (0.331 + 0.456i)17-s + (−0.998 + 0.725i)19-s + (0.763 + 0.554i)21-s + (0.269 + 0.0875i)23-s + (−0.183 − 0.0594i)27-s + (−0.106 − 0.0770i)29-s + (−1.23 + 0.896i)31-s + (−0.0597 − 0.0822i)33-s + (0.956 − 0.310i)37-s + (0.0478 − 0.147i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.296 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $0.296 - 0.955i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (649, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ 0.296 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.588161864\)
\(L(\frac12)\) \(\approx\) \(1.588161864\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.587 + 0.809i)T \)
5 \( 1 \)
good7 \( 1 - 4.32iT - 7T^{2} \)
11 \( 1 + (-0.180 + 0.555i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-0.918 + 0.298i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.36 - 1.88i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (4.35 - 3.16i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.29 - 0.419i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (0.571 + 0.415i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (6.86 - 4.98i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-5.81 + 1.89i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-3.41 - 10.5i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 7.03iT - 43T^{2} \)
47 \( 1 + (-5.33 + 7.33i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.23 - 7.20i)T + (-16.3 - 50.4i)T^{2} \)
59 \( 1 + (-2.25 - 6.93i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.48 - 4.57i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.221 - 0.304i)T + (-20.7 + 63.7i)T^{2} \)
71 \( 1 + (8.54 + 6.20i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.202 + 0.0659i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.68 + 4.12i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-9.46 - 13.0i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-4.33 + 13.3i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-9.25 + 12.7i)T + (-29.9 - 92.2i)T^{2} \)
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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

−9.372591405667987195748082258489, −8.790005753934670143663703907964, −8.196093526428841493148436964280, −7.34573726345916760053720165919, −6.01988870947908723958892305418, −5.96126568042764953289662049266, −4.66087525927444724750654690126, −3.42568884054972569208723400341, −2.51805879722013100537958692475, −1.53899596717843506836269681217, 0.60486451591521666547397604677, 2.15014098550256641390020408735, 3.48231120997483363656882528946, 4.14801052153749212844788284691, 4.90809994935911351562418974935, 6.09880165338556192206971595832, 7.14088681500507810858081519274, 7.56360114366519216140132389024, 8.602324924380434163379019239299, 9.391355647168399974142983476125

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