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Properties

Label 2-300-4.3-c4-0-12
Degree $2$
Conductor $300$
Sign $0.460 - 0.887i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−3.88 − 0.947i)2-s − 5.19i·3-s + (14.2 + 7.36i)4-s + (−4.92 + 20.1i)6-s − 12.7i·7-s + (−48.2 − 42.0i)8-s − 27·9-s + 45.0i·11-s + (38.2 − 73.8i)12-s − 1.08·13-s + (−12.1 + 49.6i)14-s + (147. + 209. i)16-s − 40.6·17-s + (104. + 25.5i)18-s − 290. i·19-s + ⋯
L(s)  = 1  + (−0.971 − 0.236i)2-s − 0.577i·3-s + (0.887 + 0.460i)4-s + (−0.136 + 0.560i)6-s − 0.260i·7-s + (−0.753 − 0.657i)8-s − 0.333·9-s + 0.372i·11-s + (0.265 − 0.512i)12-s − 0.00642·13-s + (−0.0617 + 0.253i)14-s + (0.576 + 0.817i)16-s − 0.140·17-s + (0.323 + 0.0789i)18-s − 0.804i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.460 - 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.460 - 0.887i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.460 - 0.887i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6567149075\)
\(L(\frac12)\) \(\approx\) \(0.6567149075\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (3.88 + 0.947i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 + 12.7iT - 2.40e3T^{2} \)
11 \( 1 - 45.0iT - 1.46e4T^{2} \)
13 \( 1 + 1.08T + 2.85e4T^{2} \)
17 \( 1 + 40.6T + 8.35e4T^{2} \)
19 \( 1 + 290. iT - 1.30e5T^{2} \)
23 \( 1 - 949. iT - 2.79e5T^{2} \)
29 \( 1 + 402.T + 7.07e5T^{2} \)
31 \( 1 + 762. iT - 9.23e5T^{2} \)
37 \( 1 + 1.32e3T + 1.87e6T^{2} \)
41 \( 1 + 3.20e3T + 2.82e6T^{2} \)
43 \( 1 - 2.38e3iT - 3.41e6T^{2} \)
47 \( 1 + 730. iT - 4.87e6T^{2} \)
53 \( 1 - 3.35e3T + 7.89e6T^{2} \)
59 \( 1 - 5.21e3iT - 1.21e7T^{2} \)
61 \( 1 - 4.69e3T + 1.38e7T^{2} \)
67 \( 1 + 3.35e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.48e3iT - 2.54e7T^{2} \)
73 \( 1 - 174.T + 2.83e7T^{2} \)
79 \( 1 - 1.01e4iT - 3.89e7T^{2} \)
83 \( 1 - 8.83e3iT - 4.74e7T^{2} \)
89 \( 1 + 1.93e3T + 6.27e7T^{2} \)
97 \( 1 - 1.42e4T + 8.85e7T^{2} \)
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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

−11.33528116101322285416840635101, −10.22912485436989580698677786174, −9.374022261485112179587011791234, −8.425151270088300828047185823261, −7.41185672626685850096566086297, −6.81343513467472590572989679809, −5.48315555916419902328946985256, −3.69564338780020933400162656608, −2.33084303044278753274853907928, −1.13769131781389968622586701635, 0.30299510680052331234129333340, 2.02680086276322784296045507191, 3.43050478397561655645764323839, 5.05293841918054778651701589720, 6.10796257801526103097200773529, 7.12244060709278272231824214971, 8.459724567818526250630075936179, 8.828953174183829544474233211759, 10.13415148576214026354194434771, 10.54453797031020445112463571544

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