Properties

Label 2-300-60.59-c1-0-17
Degree $2$
Conductor $300$
Sign $0.388 - 0.921i$
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 about

Normalization:  

Dirichlet series

L(s)  = 1  + (1.38 + 0.273i)2-s + (0.758 + 1.55i)3-s + (1.85 + 0.758i)4-s + (0.626 + 2.36i)6-s − 3.56·7-s + (2.36 + 1.55i)8-s + (−1.85 + 2.36i)9-s + 4.20·11-s + (0.222 + 3.45i)12-s − 2.70i·13-s + (−4.94 − 0.973i)14-s + (2.85 + 2.80i)16-s − 0.828·17-s + (−3.21 + 2.77i)18-s − 5.07i·19-s + ⋯
L(s)  = 1  + (0.981 + 0.193i)2-s + (0.437 + 0.899i)3-s + (0.925 + 0.379i)4-s + (0.255 + 0.966i)6-s − 1.34·7-s + (0.834 + 0.550i)8-s + (−0.616 + 0.787i)9-s + 1.26·11-s + (0.0642 + 0.997i)12-s − 0.749i·13-s + (−1.32 − 0.260i)14-s + (0.712 + 0.701i)16-s − 0.200·17-s + (−0.757 + 0.653i)18-s − 1.16i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.388 - 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.388 - 0.921i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.388 - 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00600 + 1.33067i\)
\(L(\frac12)\) \(\approx\) \(2.00600 + 1.33067i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.38 - 0.273i)T \)
3 \( 1 + (-0.758 - 1.55i)T \)
5 \( 1 \)
good7 \( 1 + 3.56T + 7T^{2} \)
11 \( 1 - 4.20T + 11T^{2} \)
13 \( 1 + 2.70iT - 13T^{2} \)
17 \( 1 + 0.828T + 17T^{2} \)
19 \( 1 + 5.07iT - 19T^{2} \)
23 \( 1 - 1.09iT - 23T^{2} \)
29 \( 1 + 5.55iT - 29T^{2} \)
31 \( 1 + 6.59iT - 31T^{2} \)
37 \( 1 - 5.40iT - 37T^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 - 0.531T + 43T^{2} \)
47 \( 1 + 6.22iT - 47T^{2} \)
53 \( 1 + 5.55T + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 0.701T + 61T^{2} \)
67 \( 1 - 2.04T + 67T^{2} \)
71 \( 1 + 11.3T + 71T^{2} \)
73 \( 1 + 7.70iT - 73T^{2} \)
79 \( 1 - 7.12iT - 79T^{2} \)
83 \( 1 - 3.11iT - 83T^{2} \)
89 \( 1 - 4.72iT - 89T^{2} \)
97 \( 1 + 8.10iT - 97T^{2} \)
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

−11.91885763963778460268786481638, −11.13780426168618694097339142862, −9.979971300349322671176263854534, −9.277857499937942905065502781342, −8.043489710101493088101844668593, −6.74176014595433521714950298798, −5.92258015159152407514844995384, −4.60352623886763456176305167033, −3.62812452581344428518156922242, −2.72226255833081981477679406312, 1.64303959709078520672530672139, 3.14565147961221435013572818879, 4.00548867640619918492761242357, 5.83393443555252540162556797936, 6.62466383474547701825425614472, 7.20797580207077564331214523432, 8.794036224149181509700852643137, 9.655546046383202323424093788153, 10.89018542740280362001261246653, 12.17695123055090941319069486675

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