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Properties

Label 2-300-4.3-c4-0-7
Degree $2$
Conductor $300$
Sign $-0.131 - 0.991i$
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  + (−0.264 − 3.99i)2-s + 5.19i·3-s + (−15.8 + 2.11i)4-s + (20.7 − 1.37i)6-s + 29.5i·7-s + (12.6 + 62.7i)8-s − 27·9-s − 210. i·11-s + (−10.9 − 82.4i)12-s + 135.·13-s + (118. − 7.82i)14-s + (247. − 66.9i)16-s + 7.65·17-s + (7.13 + 107. i)18-s + 166. i·19-s + ⋯
L(s)  = 1  + (−0.0661 − 0.997i)2-s + 0.577i·3-s + (−0.991 + 0.131i)4-s + (0.576 − 0.0381i)6-s + 0.603i·7-s + (0.197 + 0.980i)8-s − 0.333·9-s − 1.74i·11-s + (−0.0761 − 0.572i)12-s + 0.803·13-s + (0.602 − 0.0399i)14-s + (0.965 − 0.261i)16-s + 0.0264·17-s + (0.0220 + 0.332i)18-s + 0.461i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.131 - 0.991i$
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: \((2,\ 300,\ (\ :2),\ -0.131 - 0.991i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6172630865\)
\(L(\frac12)\) \(\approx\) \(0.6172630865\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.264 + 3.99i)T \)
3 \( 1 - 5.19iT \)
5 \( 1 \)
good7 \( 1 - 29.5iT - 2.40e3T^{2} \)
11 \( 1 + 210. iT - 1.46e4T^{2} \)
13 \( 1 - 135.T + 2.85e4T^{2} \)
17 \( 1 - 7.65T + 8.35e4T^{2} \)
19 \( 1 - 166. iT - 1.30e5T^{2} \)
23 \( 1 - 405. iT - 2.79e5T^{2} \)
29 \( 1 + 1.64e3T + 7.07e5T^{2} \)
31 \( 1 - 1.17e3iT - 9.23e5T^{2} \)
37 \( 1 + 605.T + 1.87e6T^{2} \)
41 \( 1 + 1.49e3T + 2.82e6T^{2} \)
43 \( 1 - 1.51e3iT - 3.41e6T^{2} \)
47 \( 1 - 1.88e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.95e3T + 7.89e6T^{2} \)
59 \( 1 - 500. iT - 1.21e7T^{2} \)
61 \( 1 + 928.T + 1.38e7T^{2} \)
67 \( 1 - 3.04e3iT - 2.01e7T^{2} \)
71 \( 1 + 6.96e3iT - 2.54e7T^{2} \)
73 \( 1 + 7.81e3T + 2.83e7T^{2} \)
79 \( 1 - 9.43e3iT - 3.89e7T^{2} \)
83 \( 1 - 9.48e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.37e4T + 6.27e7T^{2} \)
97 \( 1 - 7.69e3T + 8.85e7T^{2} \)
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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

−11.17736736422678807347208731950, −10.69797159648309118273134331900, −9.454518822868413696524681919561, −8.806099868134625998096695703661, −7.997804593693719311303101401033, −6.01020466345847147361460756229, −5.25991025517685933813049921862, −3.75478723574200938606432119865, −3.08203875953206036291286569155, −1.42348068707786779467077011466, 0.20255138533635836647853347664, 1.77950649512195422734803180314, 3.82783449586680509343904445245, 4.87596212808098027081545307952, 6.11535066571827272167671057345, 7.10725516269791781085368456108, 7.60961725897804876027893068984, 8.777691926480594102274219345614, 9.703451806537743808739395934843, 10.67851437829734832642465954700

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