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Properties

Label 2-1500-15.2-c1-0-23
Degree $2$
Conductor $1500$
Sign $-0.920 + 0.391i$
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  + (−1.59 + 0.678i)3-s + (−1.30 − 1.30i)7-s + (2.07 − 2.16i)9-s + 4.00i·11-s + (−0.272 + 0.272i)13-s + (2.24 − 2.24i)17-s + 3.67i·19-s + (2.95 + 1.18i)21-s + (−6.44 − 6.44i)23-s + (−1.84 + 4.85i)27-s − 2.72·29-s + 5.94·31-s + (−2.71 − 6.38i)33-s + (4.14 + 4.14i)37-s + (0.248 − 0.618i)39-s + ⋯
L(s)  = 1  + (−0.920 + 0.391i)3-s + (−0.491 − 0.491i)7-s + (0.693 − 0.720i)9-s + 1.20i·11-s + (−0.0754 + 0.0754i)13-s + (0.544 − 0.544i)17-s + 0.842i·19-s + (0.644 + 0.259i)21-s + (−1.34 − 1.34i)23-s + (−0.355 + 0.934i)27-s − 0.505·29-s + 1.06·31-s + (−0.473 − 1.11i)33-s + (0.681 + 0.681i)37-s + (0.0398 − 0.0989i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 + 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.920 + 0.391i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.920 + 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.04253284258\)
\(L(\frac12)\) \(\approx\) \(0.04253284258\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.59 - 0.678i)T \)
5 \( 1 \)
good7 \( 1 + (1.30 + 1.30i)T + 7iT^{2} \)
11 \( 1 - 4.00iT - 11T^{2} \)
13 \( 1 + (0.272 - 0.272i)T - 13iT^{2} \)
17 \( 1 + (-2.24 + 2.24i)T - 17iT^{2} \)
19 \( 1 - 3.67iT - 19T^{2} \)
23 \( 1 + (6.44 + 6.44i)T + 23iT^{2} \)
29 \( 1 + 2.72T + 29T^{2} \)
31 \( 1 - 5.94T + 31T^{2} \)
37 \( 1 + (-4.14 - 4.14i)T + 37iT^{2} \)
41 \( 1 + 0.0587iT - 41T^{2} \)
43 \( 1 + (5.58 - 5.58i)T - 43iT^{2} \)
47 \( 1 + (2.30 - 2.30i)T - 47iT^{2} \)
53 \( 1 + (6.59 + 6.59i)T + 53iT^{2} \)
59 \( 1 + 14.9T + 59T^{2} \)
61 \( 1 + 9.03T + 61T^{2} \)
67 \( 1 + (0.657 + 0.657i)T + 67iT^{2} \)
71 \( 1 + 7.42iT - 71T^{2} \)
73 \( 1 + (6.86 - 6.86i)T - 73iT^{2} \)
79 \( 1 + 0.110iT - 79T^{2} \)
83 \( 1 + (6.47 + 6.47i)T + 83iT^{2} \)
89 \( 1 + 0.461T + 89T^{2} \)
97 \( 1 + (10.1 + 10.1i)T + 97iT^{2} \)
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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

−9.646269641177558705938109072352, −8.229496130594415670482205627528, −7.40985882091074485720324890094, −6.53607822115045789410461164884, −5.98375469624705320080194012673, −4.75247238986668886380746592696, −4.31073421832772374169326826131, −3.14138769495711928460892875211, −1.60220393911516644106126985370, −0.01960655422097718726298496480, 1.43388392053503267048400029299, 2.81614906707888889095959954338, 3.90081005228605190395366978917, 5.08492310917895542520576987253, 5.98478324792674618843116477997, 6.21878791750829890267960498699, 7.44561139791810442703030970042, 8.075314688302018204701090365876, 9.098930959391947924174614616787, 9.880613514598113717174227253233

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