Properties

Label 2-300-300.11-c1-0-6
Degree $2$
Conductor $300$
Sign $-0.970 - 0.242i$
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  + (−1.39 − 0.240i)2-s + (−0.354 + 1.69i)3-s + (1.88 + 0.669i)4-s + (−0.834 + 2.07i)5-s + (0.901 − 2.27i)6-s + 4.41i·7-s + (−2.46 − 1.38i)8-s + (−2.74 − 1.20i)9-s + (1.66 − 2.69i)10-s + (1.61 − 1.17i)11-s + (−1.80 + 2.95i)12-s + (−3.38 − 2.46i)13-s + (1.06 − 6.15i)14-s + (−3.22 − 2.15i)15-s + (3.10 + 2.52i)16-s + (3.68 − 1.19i)17-s + ⋯
L(s)  = 1  + (−0.985 − 0.169i)2-s + (−0.204 + 0.978i)3-s + (0.942 + 0.334i)4-s + (−0.373 + 0.927i)5-s + (0.367 − 0.929i)6-s + 1.67i·7-s + (−0.871 − 0.489i)8-s + (−0.916 − 0.400i)9-s + (0.525 − 0.850i)10-s + (0.488 − 0.354i)11-s + (−0.520 + 0.853i)12-s + (−0.939 − 0.682i)13-s + (0.283 − 1.64i)14-s + (−0.831 − 0.555i)15-s + (0.775 + 0.630i)16-s + (0.892 − 0.290i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.970 - 0.242i$
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: \((2,\ 300,\ (\ :1/2),\ -0.970 - 0.242i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0651937 + 0.530371i\)
\(L(\frac12)\) \(\approx\) \(0.0651937 + 0.530371i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 + 0.240i)T \)
3 \( 1 + (0.354 - 1.69i)T \)
5 \( 1 + (0.834 - 2.07i)T \)
good7 \( 1 - 4.41iT - 7T^{2} \)
11 \( 1 + (-1.61 + 1.17i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (3.38 + 2.46i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-3.68 + 1.19i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.43 + 0.467i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (4.23 - 3.07i)T + (7.10 - 21.8i)T^{2} \)
29 \( 1 + (0.261 + 0.0850i)T + (23.4 + 17.0i)T^{2} \)
31 \( 1 + (7.27 - 2.36i)T + (25.0 - 18.2i)T^{2} \)
37 \( 1 + (-6.29 - 4.57i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.40 - 1.92i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 - 8.38iT - 43T^{2} \)
47 \( 1 + (0.337 - 1.03i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (2.15 + 0.698i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.92 - 5.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-11.9 + 8.69i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-3.58 + 1.16i)T + (54.2 - 39.3i)T^{2} \)
71 \( 1 + (2.48 - 7.64i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (3.17 - 2.30i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.47 + 1.12i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (3.08 + 9.50i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + (-1.36 - 1.87i)T + (-27.5 + 84.6i)T^{2} \)
97 \( 1 + (-0.529 + 1.63i)T + (-78.4 - 57.0i)T^{2} \)
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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

−11.64573973464256017771493034230, −11.34646710657367796653437038871, −10.01859507124322657788933613072, −9.613146906095494174535181763808, −8.549054962730227043036008863416, −7.63883901200250355948829471112, −6.24104126197033150578682612056, −5.40611075233045512360474794629, −3.43636544934652488338064805071, −2.59977900806586609847137239165, 0.54238777283537274730157548759, 1.80647472137822831437867299780, 3.99998581936395147351815806768, 5.53165339377895112682335866381, 6.91420346152738784801412712020, 7.44709303305742306919479852867, 8.202937862770551234916712515224, 9.380698320456009827778035911833, 10.26876064024748760634100480916, 11.37033726935785846667103790039

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