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Properties

Label 2-300-75.2-c1-0-0
Degree $2$
Conductor $300$
Sign $-0.995 + 0.0919i$
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  + (−1.21 + 1.23i)3-s + (−1.48 + 1.67i)5-s + (−0.245 − 0.245i)7-s + (−0.0284 − 2.99i)9-s + (0.879 + 1.21i)11-s + (−5.29 + 0.839i)13-s + (−0.248 − 3.86i)15-s + (−6.61 − 3.36i)17-s + (−3.22 + 1.04i)19-s + (0.601 − 0.00285i)21-s + (1.61 + 0.255i)23-s + (−0.592 − 4.96i)25-s + (3.72 + 3.62i)27-s + (0.637 − 1.96i)29-s + (2.11 + 6.51i)31-s + ⋯
L(s)  = 1  + (−0.703 + 0.710i)3-s + (−0.663 + 0.747i)5-s + (−0.0927 − 0.0927i)7-s + (−0.00949 − 0.999i)9-s + (0.265 + 0.365i)11-s + (−1.46 + 0.232i)13-s + (−0.0640 − 0.997i)15-s + (−1.60 − 0.817i)17-s + (−0.738 + 0.240i)19-s + (0.131 − 0.000622i)21-s + (0.335 + 0.0531i)23-s + (−0.118 − 0.992i)25-s + (0.717 + 0.696i)27-s + (0.118 − 0.364i)29-s + (0.380 + 1.17i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.995 + 0.0919i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (77, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.995 + 0.0919i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0142561 - 0.309459i\)
\(L(\frac12)\) \(\approx\) \(0.0142561 - 0.309459i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (1.21 - 1.23i)T \)
5 \( 1 + (1.48 - 1.67i)T \)
good7 \( 1 + (0.245 + 0.245i)T + 7iT^{2} \)
11 \( 1 + (-0.879 - 1.21i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (5.29 - 0.839i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (6.61 + 3.36i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (3.22 - 1.04i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (-1.61 - 0.255i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (-0.637 + 1.96i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.11 - 6.51i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.18 - 7.50i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (2.49 - 3.44i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.03 - 3.03i)T - 43iT^{2} \)
47 \( 1 + (-1.43 - 2.81i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-0.212 + 0.108i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (-2.57 - 1.87i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.40 + 2.47i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (5.71 - 11.2i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (-15.0 - 4.87i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.02 - 12.7i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (9.94 + 3.23i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-6.61 + 12.9i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (10.2 - 7.42i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.73 + 1.39i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.84607360108972278433040524133, −11.40510958956262967058675164436, −10.35478473747249875207310379523, −9.695257250590075364483369388526, −8.521510403445102355911784175649, −7.03029840644018681017421878147, −6.59703741679422946438162034174, −4.93495123534893596415362431017, −4.21443329205651174784902776049, −2.73967819796990053805515169844, 0.23399507676528097136549829016, 2.19431252345128773696506683200, 4.22264465875412156071635776244, 5.16885776911318391689631766988, 6.37847025958999248395765125495, 7.33908034836031097971120386014, 8.294788162893822303282138457678, 9.216197110385370631373424249457, 10.63090329667969763544852144929, 11.37276610564793305526163071199

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