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Properties

Label 2-300-5.3-c8-0-0
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 + (−1.93e3 + 1.93e3i)7-s + 2.18e3i·9-s + 3.22e3·11-s + (−2.54e4 − 2.54e4i)13-s + (9.82e4 − 9.82e4i)17-s + 1.08e5i·19-s + 1.28e5·21-s + (2.78e5 + 2.78e5i)23-s + (7.23e4 − 7.23e4i)27-s + 1.08e6i·29-s − 9.41e5·31-s + (−1.06e5 − 1.06e5i)33-s + (2.45e6 − 2.45e6i)37-s + 1.68e6i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (−0.807 + 0.807i)7-s + 0.333i·9-s + 0.220·11-s + (−0.891 − 0.891i)13-s + (1.17 − 1.17i)17-s + 0.831i·19-s + 0.659·21-s + (0.994 + 0.994i)23-s + (0.136 − 0.136i)27-s + 1.54i·29-s − 1.01·31-s + (−0.0899 − 0.0899i)33-s + (1.31 − 1.31i)37-s + 0.727i·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

\(L(\frac{9}{2})\) \(\approx\) \(0.08365172662\)
\(L(\frac12)\) \(\approx\) \(0.08365172662\)
\(L(5)\) 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 \( 1 + (1.93e3 - 1.93e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 3.22e3T + 2.14e8T^{2} \)
13 \( 1 + (2.54e4 + 2.54e4i)T + 8.15e8iT^{2} \)
17 \( 1 + (-9.82e4 + 9.82e4i)T - 6.97e9iT^{2} \)
19 \( 1 - 1.08e5iT - 1.69e10T^{2} \)
23 \( 1 + (-2.78e5 - 2.78e5i)T + 7.83e10iT^{2} \)
29 \( 1 - 1.08e6iT - 5.00e11T^{2} \)
31 \( 1 + 9.41e5T + 8.52e11T^{2} \)
37 \( 1 + (-2.45e6 + 2.45e6i)T - 3.51e12iT^{2} \)
41 \( 1 - 2.41e6T + 7.98e12T^{2} \)
43 \( 1 + (3.27e6 + 3.27e6i)T + 1.16e13iT^{2} \)
47 \( 1 + (-2.76e6 + 2.76e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (1.44e6 + 1.44e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.34e5iT - 1.46e14T^{2} \)
61 \( 1 - 6.73e6T + 1.91e14T^{2} \)
67 \( 1 + (2.13e7 - 2.13e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 7.79e6T + 6.45e14T^{2} \)
73 \( 1 + (5.30e6 + 5.30e6i)T + 8.06e14iT^{2} \)
79 \( 1 + 3.57e7iT - 1.51e15T^{2} \)
83 \( 1 + (-2.00e7 - 2.00e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 5.88e7iT - 3.93e15T^{2} \)
97 \( 1 + (6.71e6 - 6.71e6i)T - 7.83e15iT^{2} \)
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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

−10.80006807688005642090363897294, −9.748321182360060062879668940893, −9.073001475403948237406969574258, −7.67719075792377877562808710582, −7.03750000302108902536868916751, −5.66314596348546672294631242783, −5.28531057689074733078041860452, −3.46848477547563245483885153738, −2.57001292813904585956176514088, −1.13300909447270509708568965429, 0.02151888259604066918583419361, 1.12403650109249900909594914682, 2.74188848142007947890947363051, 3.93723601124325192378555814971, 4.73965866183423246145817451383, 6.10622943554217071597066266881, 6.84694627728042470392182486291, 7.906830689008385536060065721766, 9.289252606063450007888384292274, 9.887318028356683320629119491363

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