Properties

Label 2-300-300.191-c1-0-26
Degree $2$
Conductor $300$
Sign $-0.00842 - 0.999i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.24 + 0.671i)2-s + (−0.339 + 1.69i)3-s + (1.09 + 1.67i)4-s + (2.10 − 0.759i)5-s + (−1.56 + 1.88i)6-s + 0.738i·7-s + (0.244 + 2.81i)8-s + (−2.76 − 1.15i)9-s + (3.12 + 0.467i)10-s + (−1.63 − 1.18i)11-s + (−3.21 + 1.29i)12-s + (3.83 − 2.78i)13-s + (−0.496 + 0.919i)14-s + (0.574 + 3.83i)15-s + (−1.58 + 3.67i)16-s + (−3.35 − 1.08i)17-s + ⋯
L(s)  = 1  + (0.880 + 0.474i)2-s + (−0.196 + 0.980i)3-s + (0.549 + 0.835i)4-s + (0.940 − 0.339i)5-s + (−0.638 + 0.769i)6-s + 0.279i·7-s + (0.0863 + 0.996i)8-s + (−0.923 − 0.384i)9-s + (0.989 + 0.147i)10-s + (−0.491 − 0.357i)11-s + (−0.927 + 0.374i)12-s + (1.06 − 0.773i)13-s + (−0.132 + 0.245i)14-s + (0.148 + 0.988i)15-s + (−0.397 + 0.917i)16-s + (−0.813 − 0.264i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00842 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00842 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.00842 - 0.999i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.00842 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53227 + 1.54524i\)
\(L(\frac12)\) \(\approx\) \(1.53227 + 1.54524i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.24 - 0.671i)T \)
3 \( 1 + (0.339 - 1.69i)T \)
5 \( 1 + (-2.10 + 0.759i)T \)
good7 \( 1 - 0.738iT - 7T^{2} \)
11 \( 1 + (1.63 + 1.18i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-3.83 + 2.78i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (3.35 + 1.08i)T + (13.7 + 9.99i)T^{2} \)
19 \( 1 + (2.82 + 0.919i)T + (15.3 + 11.1i)T^{2} \)
23 \( 1 + (-1.01 - 0.736i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (7.24 - 2.35i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.78 - 2.20i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.994 - 0.722i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (1.02 + 1.40i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 2.44iT - 43T^{2} \)
47 \( 1 + (3.13 + 9.65i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-11.6 + 3.80i)T + (42.8 - 31.1i)T^{2} \)
59 \( 1 + (3.97 - 2.88i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-5.09 - 3.70i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + (-8.81 - 2.86i)T + (54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.32 + 13.3i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (4.33 + 3.14i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (14.7 - 4.79i)T + (63.9 - 46.4i)T^{2} \)
83 \( 1 + (0.885 - 2.72i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 + (7.36 - 10.1i)T + (-27.5 - 84.6i)T^{2} \)
97 \( 1 + (-2.66 - 8.19i)T + (-78.4 + 57.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.03538757712295367368168401628, −10.99719902685529423363347210495, −10.33310376567270796332013477261, −8.935499961186408998778609535538, −8.402404244818238592730909346698, −6.68224239898207975714524726792, −5.69401855216671253833913124204, −5.14734456308689625734190345298, −3.86520950810232097671242736254, −2.58866319656563290060974032905, 1.60607645968976556523048958037, 2.62116481595945091731760777629, 4.27839987174814066418378132354, 5.70215985742012596814317036519, 6.37510840913439939833928174309, 7.19870905926358002409692628334, 8.686465746543603770065505108131, 9.932537494854582851838304499018, 10.92907941351706008522564323192, 11.46648445430250210977533667784

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