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Properties

Label 2-1500-25.19-c1-0-2
Degree $2$
Conductor $1500$
Sign $-0.920 - 0.390i$
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 + 3.80i·7-s + (−0.309 + 0.951i)9-s + (0.0589 + 0.181i)11-s + (−1.59 − 0.518i)13-s + (−2.70 + 3.72i)17-s + (−2.13 − 1.55i)19-s + (−3.08 + 2.23i)21-s + (−6.04 + 1.96i)23-s + (−0.951 + 0.309i)27-s + (2.03 − 1.48i)29-s + (−3.03 − 2.20i)31-s + (−0.112 + 0.154i)33-s + (11.2 + 3.66i)37-s + (−0.518 − 1.59i)39-s + ⋯
L(s)  = 1  + (0.339 + 0.467i)3-s + 1.44i·7-s + (−0.103 + 0.317i)9-s + (0.0177 + 0.0546i)11-s + (−0.442 − 0.143i)13-s + (−0.656 + 0.903i)17-s + (−0.490 − 0.356i)19-s + (−0.672 + 0.488i)21-s + (−1.26 + 0.409i)23-s + (−0.183 + 0.0594i)27-s + (0.378 − 0.275i)29-s + (−0.544 − 0.395i)31-s + (−0.0195 + 0.0268i)33-s + (1.85 + 0.602i)37-s + (−0.0831 − 0.255i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1500 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.390i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1500\)    =    \(2^{2} \cdot 3 \cdot 5^{3}\)
Sign: $-0.920 - 0.390i$
Analytic conductor: \(11.9775\)
Root analytic conductor: \(3.46086\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1500} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1500,\ (\ :1/2),\ -0.920 - 0.390i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.109040322\)
\(L(\frac12)\) \(\approx\) \(1.109040322\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.80iT - 7T^{2} \)
11 \( 1 + (-0.0589 - 0.181i)T + (-8.89 + 6.46i)T^{2} \)
13 \( 1 + (1.59 + 0.518i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.70 - 3.72i)T + (-5.25 - 16.1i)T^{2} \)
19 \( 1 + (2.13 + 1.55i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (6.04 - 1.96i)T + (18.6 - 13.5i)T^{2} \)
29 \( 1 + (-2.03 + 1.48i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (3.03 + 2.20i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-11.2 - 3.66i)T + (29.9 + 21.7i)T^{2} \)
41 \( 1 + (2.22 - 6.83i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 9.22iT - 43T^{2} \)
47 \( 1 + (2.67 + 3.67i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (5.54 + 7.62i)T + (-16.3 + 50.4i)T^{2} \)
59 \( 1 + (2.20 - 6.79i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-2.94 - 9.06i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.55 - 4.89i)T + (-20.7 - 63.7i)T^{2} \)
71 \( 1 + (10.7 - 7.81i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-4.95 + 1.61i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (2.51 - 1.82i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.74 + 3.78i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (4.30 + 13.2i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (-3.93 - 5.41i)T + (-29.9 + 92.2i)T^{2} \)
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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

−9.796389624371293126012998083484, −8.939291259983501904734162726280, −8.438830554887285686952708964332, −7.63493280454381862628919224961, −6.36609116478488142573734671154, −5.77415363048354870278927340697, −4.79462394615740301526426872557, −3.92015173349295031479006243093, −2.70734052740533936532929851542, −1.99001347376298311678419677199, 0.39189667367804575852935373216, 1.77170073835892279768341189541, 2.94824780576456856718870202014, 4.07495287647846956957331051957, 4.70496148368413124968436531522, 6.09789638806125623118797493427, 6.80627598928712803689122440632, 7.58252908805505708312924156807, 8.097815255216233850965338995053, 9.212402489931489968394929882567

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