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Properties

Label 2-300-5.3-c8-0-5
Degree $2$
Conductor $300$
Sign $-0.973 - 0.229i$
Analytic cond. $122.213$
Root an. cond. $11.0550$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−33.0 − 33.0i)3-s + (−2.66e3 + 2.66e3i)7-s + 2.18e3i·9-s − 4.23e3·11-s + (3.30e4 + 3.30e4i)13-s + (−2.76e4 + 2.76e4i)17-s + 3.57e4i·19-s + 1.76e5·21-s + (1.43e5 + 1.43e5i)23-s + (7.23e4 − 7.23e4i)27-s + 4.15e4i·29-s + 1.33e6·31-s + (1.39e5 + 1.39e5i)33-s + (−2.05e5 + 2.05e5i)37-s − 2.18e6i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−1.11 + 1.11i)7-s + 0.333i·9-s − 0.288·11-s + (1.15 + 1.15i)13-s + (−0.331 + 0.331i)17-s + 0.274i·19-s + 0.906·21-s + (0.513 + 0.513i)23-s + (0.136 − 0.136i)27-s + 0.0588i·29-s + 1.44·31-s + (0.117 + 0.117i)33-s + (−0.109 + 0.109i)37-s − 0.945i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.973 - 0.229i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ -0.973 - 0.229i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.8086877809\)
\(L(\frac12)\) \(\approx\) \(0.8086877809\)
\(L(5)\) 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 \( 1 + (33.0 + 33.0i)T \)
5 \( 1 \)
good7 \( 1 + (2.66e3 - 2.66e3i)T - 5.76e6iT^{2} \)
11 \( 1 + 4.23e3T + 2.14e8T^{2} \)
13 \( 1 + (-3.30e4 - 3.30e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (2.76e4 - 2.76e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 3.57e4iT - 1.69e10T^{2} \)
23 \( 1 + (-1.43e5 - 1.43e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 4.15e4iT - 5.00e11T^{2} \)
31 \( 1 - 1.33e6T + 8.52e11T^{2} \)
37 \( 1 + (2.05e5 - 2.05e5i)T - 3.51e12iT^{2} \)
41 \( 1 - 6.81e5T + 7.98e12T^{2} \)
43 \( 1 + (-1.31e6 - 1.31e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (5.69e6 - 5.69e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (8.35e6 + 8.35e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 8.05e6iT - 1.46e14T^{2} \)
61 \( 1 + 8.53e6T + 1.91e14T^{2} \)
67 \( 1 + (-2.55e7 + 2.55e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 1.98e7T + 6.45e14T^{2} \)
73 \( 1 + (1.32e7 + 1.32e7i)T + 8.06e14iT^{2} \)
79 \( 1 - 2.86e7iT - 1.51e15T^{2} \)
83 \( 1 + (-4.35e7 - 4.35e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 1.06e8iT - 3.93e15T^{2} \)
97 \( 1 + (-8.59e7 + 8.59e7i)T - 7.83e15iT^{2} \)
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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

−10.95343397049800295608852192221, −9.703835029625539400961721571025, −8.965073975669837569401659485815, −7.982188009039579745316149652864, −6.44112790453860016333586273351, −6.29406084547241973034940805885, −4.98601443752588811799053071507, −3.58917600187064883692569316447, −2.44796274680594280656290650553, −1.26575307372709982282899536206, 0.22314784991820490975707546590, 0.937873856905661595028289915752, 2.90939480391433543104164758766, 3.74248395597680257121841583577, 4.85300658176455321495173354263, 6.08556954573065174493976052324, 6.81662265943681009880285566947, 7.978062601756345578989929963030, 9.103607481135312292821274054226, 10.20698567409388074991752811535

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