Properties

Label 2-300-100.67-c1-0-29
Degree $2$
Conductor $300$
Sign $-0.993 - 0.110i$
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  + (0.170 − 1.40i)2-s + (−0.987 − 0.156i)3-s + (−1.94 − 0.477i)4-s + (0.0997 − 2.23i)5-s + (−0.387 + 1.36i)6-s + (2.03 − 2.03i)7-s + (−1.00 + 2.64i)8-s + (0.951 + 0.309i)9-s + (−3.11 − 0.520i)10-s + (−3.90 + 1.27i)11-s + (1.84 + 0.775i)12-s + (−2.28 − 4.48i)13-s + (−2.50 − 3.19i)14-s + (−0.447 + 2.19i)15-s + (3.54 + 1.85i)16-s + (1.11 + 7.01i)17-s + ⋯
L(s)  = 1  + (0.120 − 0.992i)2-s + (−0.570 − 0.0903i)3-s + (−0.971 − 0.238i)4-s + (0.0445 − 0.999i)5-s + (−0.158 + 0.555i)6-s + (0.767 − 0.767i)7-s + (−0.354 + 0.935i)8-s + (0.317 + 0.103i)9-s + (−0.986 − 0.164i)10-s + (−1.17 + 0.382i)11-s + (0.532 + 0.223i)12-s + (−0.633 − 1.24i)13-s + (−0.669 − 0.854i)14-s + (−0.115 + 0.565i)15-s + (0.885 + 0.464i)16-s + (0.269 + 1.70i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 - 0.110i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.993 - 0.110i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (67, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ -0.993 - 0.110i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0451046 + 0.814785i\)
\(L(\frac12)\) \(\approx\) \(0.0451046 + 0.814785i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.170 + 1.40i)T \)
3 \( 1 + (0.987 + 0.156i)T \)
5 \( 1 + (-0.0997 + 2.23i)T \)
good7 \( 1 + (-2.03 + 2.03i)T - 7iT^{2} \)
11 \( 1 + (3.90 - 1.27i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.28 + 4.48i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (-1.11 - 7.01i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (5.34 + 3.88i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-0.683 + 1.34i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-0.370 - 0.509i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-3.16 + 4.35i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-8.74 + 4.45i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (-1.20 + 3.69i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (-4.26 - 4.26i)T + 43iT^{2} \)
47 \( 1 + (-1.30 + 8.24i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.811 - 5.12i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-1.42 + 4.38i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.83 + 8.72i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-11.3 + 1.80i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (3.24 + 4.45i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-2.13 - 1.08i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (3.03 - 2.20i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.25 - 7.89i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-6.30 + 2.04i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (4.60 + 0.728i)T + (92.2 + 29.9i)T^{2} \)
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

−11.02612596392752317646544509277, −10.61373095637986092409970439477, −9.732548900399950907778975092390, −8.319870620036748065409218858626, −7.79503823782123662278443092281, −5.83918776244529458261830364488, −4.89993315560062455989340313006, −4.18414918492297991643654633799, −2.21623695780957568139312179235, −0.62144602921643518019967866072, 2.65460357205489828905618140328, 4.45578137443363754721362819720, 5.37343776091710697797950404463, 6.32519498762305167323609000057, 7.29518955510853866669555088954, 8.147261358114900629887887938768, 9.354006430752177598794958564089, 10.28698712738544554495659512411, 11.40671337953209995871598589199, 12.10218163864274262399885531360

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