Properties

Label 2-300-300.11-c1-0-26
Degree $2$
Conductor $300$
Sign $0.983 - 0.179i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.38 − 0.284i)2-s + (1.64 + 0.556i)3-s + (1.83 + 0.788i)4-s + (1.45 + 1.69i)5-s + (−2.11 − 1.23i)6-s − 2.78i·7-s + (−2.32 − 1.61i)8-s + (2.38 + 1.82i)9-s + (−1.53 − 2.76i)10-s + (3.44 − 2.50i)11-s + (2.57 + 2.31i)12-s + (0.188 + 0.137i)13-s + (−0.792 + 3.85i)14-s + (1.43 + 3.59i)15-s + (2.75 + 2.89i)16-s + (−2.43 + 0.790i)17-s + ⋯
L(s)  = 1  + (−0.979 − 0.201i)2-s + (0.946 + 0.321i)3-s + (0.918 + 0.394i)4-s + (0.650 + 0.759i)5-s + (−0.862 − 0.505i)6-s − 1.05i·7-s + (−0.820 − 0.571i)8-s + (0.793 + 0.608i)9-s + (−0.484 − 0.875i)10-s + (1.03 − 0.755i)11-s + (0.743 + 0.668i)12-s + (0.0524 + 0.0380i)13-s + (−0.211 + 1.03i)14-s + (0.371 + 0.928i)15-s + (0.689 + 0.724i)16-s + (−0.589 + 0.191i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 - 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.983 - 0.179i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.983 - 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31129 + 0.118771i\)
\(L(\frac12)\) \(\approx\) \(1.31129 + 0.118771i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.38 + 0.284i)T \)
3 \( 1 + (-1.64 - 0.556i)T \)
5 \( 1 + (-1.45 - 1.69i)T \)
good7 \( 1 + 2.78iT - 7T^{2} \)
11 \( 1 + (-3.44 + 2.50i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (-0.188 - 0.137i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (2.43 - 0.790i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (7.47 - 2.42i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-2.43 + 1.76i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (-5.68 - 1.84i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.25 + 0.732i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.01 + 2.19i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (6.28 - 8.65i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.67iT - 43T^{2} \)
47 \( 1 + (-0.254 + 0.783i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (9.82 + 3.19i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (3.95 + 2.87i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-9.01 + 6.54i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (9.48 - 3.08i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (1.64 - 5.07i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (4.15 - 3.01i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (7.14 + 2.32i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.275 + 0.847i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (0.121 + 0.167i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.309 + 0.953i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.26433081514657586278780216764, −10.53542685818491692767338614166, −9.998945170801601650417245359663, −8.900969426855209410022369015703, −8.261057771867767469302923586118, −6.95085357642238998180155629831, −6.41721973937076813313748373154, −4.12780723351465430119829737368, −3.07870876154015002387416156240, −1.69602826959923956169121473931, 1.61794785702663973526558513907, 2.57427434759427039536961945003, 4.61365124109493336453252488830, 6.18286760940898696843096106300, 6.90307529280821941627412485731, 8.323805096919804699711328613234, 8.908807545611243434846416708557, 9.377286819843423928752283213571, 10.40441533627389381445625977613, 11.88000986184235300400708905060

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