Properties

Label 2-300-100.67-c1-0-20
Degree $2$
Conductor $300$
Sign $0.205 + 0.978i$
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  + (−0.344 − 1.37i)2-s + (0.987 + 0.156i)3-s + (−1.76 + 0.944i)4-s + (1.92 − 1.13i)5-s + (−0.125 − 1.40i)6-s + (0.176 − 0.176i)7-s + (1.90 + 2.09i)8-s + (0.951 + 0.309i)9-s + (−2.22 − 2.25i)10-s + (2.50 − 0.814i)11-s + (−1.88 + 0.656i)12-s + (−0.0867 − 0.170i)13-s + (−0.303 − 0.181i)14-s + (2.07 − 0.821i)15-s + (2.21 − 3.32i)16-s + (−0.150 − 0.952i)17-s + ⋯
L(s)  = 1  + (−0.243 − 0.969i)2-s + (0.570 + 0.0903i)3-s + (−0.881 + 0.472i)4-s + (0.861 − 0.508i)5-s + (−0.0511 − 0.575i)6-s + (0.0667 − 0.0667i)7-s + (0.672 + 0.740i)8-s + (0.317 + 0.103i)9-s + (−0.702 − 0.711i)10-s + (0.755 − 0.245i)11-s + (−0.545 + 0.189i)12-s + (−0.0240 − 0.0472i)13-s + (−0.0810 − 0.0485i)14-s + (0.537 − 0.212i)15-s + (0.554 − 0.832i)16-s + (−0.0366 − 0.231i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.205 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.205 + 0.978i$
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: \((2,\ 300,\ (\ :1/2),\ 0.205 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14989 - 0.933374i\)
\(L(\frac12)\) \(\approx\) \(1.14989 - 0.933374i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.344 + 1.37i)T \)
3 \( 1 + (-0.987 - 0.156i)T \)
5 \( 1 + (-1.92 + 1.13i)T \)
good7 \( 1 + (-0.176 + 0.176i)T - 7iT^{2} \)
11 \( 1 + (-2.50 + 0.814i)T + (8.89 - 6.46i)T^{2} \)
13 \( 1 + (0.0867 + 0.170i)T + (-7.64 + 10.5i)T^{2} \)
17 \( 1 + (0.150 + 0.952i)T + (-16.1 + 5.25i)T^{2} \)
19 \( 1 + (3.11 + 2.26i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.11 - 2.17i)T + (-13.5 - 18.6i)T^{2} \)
29 \( 1 + (-2.72 - 3.74i)T + (-8.96 + 27.5i)T^{2} \)
31 \( 1 + (-4.34 + 5.98i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (10.3 - 5.26i)T + (21.7 - 29.9i)T^{2} \)
41 \( 1 + (1.65 - 5.07i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + (6.04 + 6.04i)T + 43iT^{2} \)
47 \( 1 + (0.924 - 5.83i)T + (-44.6 - 14.5i)T^{2} \)
53 \( 1 + (-0.172 + 1.08i)T + (-50.4 - 16.3i)T^{2} \)
59 \( 1 + (-0.0901 + 0.277i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (-3.76 - 11.5i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-10.2 + 1.63i)T + (63.7 - 20.7i)T^{2} \)
71 \( 1 + (-5.12 - 7.05i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.09 + 2.08i)T + (42.9 + 59.0i)T^{2} \)
79 \( 1 + (9.57 - 6.96i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (-1.04 - 6.62i)T + (-78.9 + 25.6i)T^{2} \)
89 \( 1 + (-13.6 + 4.43i)T + (72.0 - 52.3i)T^{2} \)
97 \( 1 + (4.05 + 0.642i)T + (92.2 + 29.9i)T^{2} \)
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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

−11.54956574214160140638862334663, −10.42998793231079459562974837807, −9.677215526856318597270105022438, −8.889281641893929879796100201668, −8.207353178512942026048306989928, −6.68180190996068460865385184306, −5.18424139335253249705077119078, −4.10083283879691465382226019155, −2.73490860034698432858757229059, −1.42524349118679670956892229707, 1.86462877299126340652844465954, 3.71339399933995519655728488636, 5.09044558987695188518935616717, 6.37506358406533233925336395436, 6.89349381407798850771895607685, 8.210427092445792543744917052017, 8.939602618216256456342686205364, 9.926503270580096076801271442736, 10.53952037466730206773745767705, 12.14869341027338884061847072184

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