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Properties

Label 2-300-5.3-c8-0-15
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 + (−2.57e3 + 2.57e3i)7-s + 2.18e3i·9-s + 1.03e4·11-s + (−48.7 − 48.7i)13-s + (1.88e4 − 1.88e4i)17-s − 2.13e5i·19-s − 1.70e5·21-s + (−9.05e4 − 9.05e4i)23-s + (−7.23e4 + 7.23e4i)27-s − 5.93e5i·29-s − 1.46e6·31-s + (3.40e5 + 3.40e5i)33-s + (1.26e5 − 1.26e5i)37-s − 3.22e3i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (−1.07 + 1.07i)7-s + 0.333i·9-s + 0.704·11-s + (−0.00170 − 0.00170i)13-s + (0.226 − 0.226i)17-s − 1.63i·19-s − 0.877·21-s + (−0.323 − 0.323i)23-s + (−0.136 + 0.136i)27-s − 0.838i·29-s − 1.58·31-s + (0.287 + 0.287i)33-s + (0.0672 − 0.0672i)37-s − 0.00139i·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

\(L(\frac{9}{2})\) \(\approx\) \(1.850182904\)
\(L(\frac12)\) \(\approx\) \(1.850182904\)
\(L(5)\) 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 \( 1 + (2.57e3 - 2.57e3i)T - 5.76e6iT^{2} \)
11 \( 1 - 1.03e4T + 2.14e8T^{2} \)
13 \( 1 + (48.7 + 48.7i)T + 8.15e8iT^{2} \)
17 \( 1 + (-1.88e4 + 1.88e4i)T - 6.97e9iT^{2} \)
19 \( 1 + 2.13e5iT - 1.69e10T^{2} \)
23 \( 1 + (9.05e4 + 9.05e4i)T + 7.83e10iT^{2} \)
29 \( 1 + 5.93e5iT - 5.00e11T^{2} \)
31 \( 1 + 1.46e6T + 8.52e11T^{2} \)
37 \( 1 + (-1.26e5 + 1.26e5i)T - 3.51e12iT^{2} \)
41 \( 1 + 4.39e5T + 7.98e12T^{2} \)
43 \( 1 + (-9.67e5 - 9.67e5i)T + 1.16e13iT^{2} \)
47 \( 1 + (-2.57e6 + 2.57e6i)T - 2.38e13iT^{2} \)
53 \( 1 + (-7.87e6 - 7.87e6i)T + 6.22e13iT^{2} \)
59 \( 1 - 1.92e7iT - 1.46e14T^{2} \)
61 \( 1 - 4.73e6T + 1.91e14T^{2} \)
67 \( 1 + (-1.13e7 + 1.13e7i)T - 4.06e14iT^{2} \)
71 \( 1 - 4.70e7T + 6.45e14T^{2} \)
73 \( 1 + (1.10e7 + 1.10e7i)T + 8.06e14iT^{2} \)
79 \( 1 + 5.16e7iT - 1.51e15T^{2} \)
83 \( 1 + (6.05e7 + 6.05e7i)T + 2.25e15iT^{2} \)
89 \( 1 - 5.57e7iT - 3.93e15T^{2} \)
97 \( 1 + (5.81e7 - 5.81e7i)T - 7.83e15iT^{2} \)
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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

−10.09919304338509637932986456214, −9.153598520850348207289448290752, −8.872926252950609364885925010428, −7.39619654651816683831388208929, −6.37030045198769344482978175792, −5.41000348869648853995937169552, −4.13779624873565697850059005213, −3.04766688563184102170763720760, −2.19363908157530852505294740252, −0.46094714595421779977141652114, 0.808563843609518263756074386382, 1.86270908089322276099547480468, 3.47868724771176539973626277460, 3.88203183434218829701140919743, 5.65134080858458639382616482063, 6.68292740276148544965141692632, 7.38393509372468949089916589831, 8.434312836654723122913408060231, 9.561011124087369796330563140530, 10.17846417915503585357766387439

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