Properties

Label 2-300-75.2-c1-0-5
Degree $2$
Conductor $300$
Sign $0.837 - 0.546i$
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  + (1.71 + 0.227i)3-s + (−0.0496 + 2.23i)5-s + (−2.02 − 2.02i)7-s + (2.89 + 0.779i)9-s + (2.82 + 3.89i)11-s + (5.74 − 0.910i)13-s + (−0.593 + 3.82i)15-s + (−1.36 − 0.694i)17-s + (−3.53 + 1.14i)19-s + (−3.01 − 3.92i)21-s + (−2.52 − 0.400i)23-s + (−4.99 − 0.222i)25-s + (4.79 + 1.99i)27-s + (−0.0334 + 0.102i)29-s + (−3.13 − 9.63i)31-s + ⋯
L(s)  = 1  + (0.991 + 0.131i)3-s + (−0.0222 + 0.999i)5-s + (−0.763 − 0.763i)7-s + (0.965 + 0.259i)9-s + (0.852 + 1.17i)11-s + (1.59 − 0.252i)13-s + (−0.153 + 0.988i)15-s + (−0.330 − 0.168i)17-s + (−0.810 + 0.263i)19-s + (−0.656 − 0.857i)21-s + (−0.526 − 0.0834i)23-s + (−0.999 − 0.0444i)25-s + (0.923 + 0.384i)27-s + (−0.00620 + 0.0191i)29-s + (−0.562 − 1.73i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.837 - 0.546i$
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: \((2,\ 300,\ (\ :1/2),\ 0.837 - 0.546i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67217 + 0.497228i\)
\(L(\frac12)\) \(\approx\) \(1.67217 + 0.497228i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.71 - 0.227i)T \)
5 \( 1 + (0.0496 - 2.23i)T \)
good7 \( 1 + (2.02 + 2.02i)T + 7iT^{2} \)
11 \( 1 + (-2.82 - 3.89i)T + (-3.39 + 10.4i)T^{2} \)
13 \( 1 + (-5.74 + 0.910i)T + (12.3 - 4.01i)T^{2} \)
17 \( 1 + (1.36 + 0.694i)T + (9.99 + 13.7i)T^{2} \)
19 \( 1 + (3.53 - 1.14i)T + (15.3 - 11.1i)T^{2} \)
23 \( 1 + (2.52 + 0.400i)T + (21.8 + 7.10i)T^{2} \)
29 \( 1 + (0.0334 - 0.102i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.13 + 9.63i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.418 + 2.64i)T + (-35.1 + 11.4i)T^{2} \)
41 \( 1 + (2.26 - 3.11i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (1.86 - 1.86i)T - 43iT^{2} \)
47 \( 1 + (4.48 + 8.80i)T + (-27.6 + 38.0i)T^{2} \)
53 \( 1 + (-9.81 + 4.99i)T + (31.1 - 42.8i)T^{2} \)
59 \( 1 + (0.210 + 0.153i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (4.36 - 3.17i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.74 + 3.42i)T + (-39.3 - 54.2i)T^{2} \)
71 \( 1 + (14.1 + 4.61i)T + (57.4 + 41.7i)T^{2} \)
73 \( 1 + (0.320 - 2.02i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-8.17 - 2.65i)T + (63.9 + 46.4i)T^{2} \)
83 \( 1 + (-3.16 + 6.21i)T + (-48.7 - 67.1i)T^{2} \)
89 \( 1 + (3.84 - 2.79i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-14.9 + 7.63i)T + (57.0 - 78.4i)T^{2} \)
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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

−11.76042391233688950546089484067, −10.60458783011223595056924770595, −10.00178789895850042419016600362, −9.101612004162593700464709705806, −7.939296879834180061333935452239, −6.96882580621110932870094957801, −6.28649734898229651008684514739, −4.09747152473321876037296183093, −3.58109729191853120962741000407, −2.03735735037457961391666708588, 1.49380992192838559995908722478, 3.23285419006640877930487140841, 4.16951277208067828121087715997, 5.84107971486920625158846600252, 6.65517915734473633920661597145, 8.343081392233940415514081968919, 8.771628406400580532211816494278, 9.296424844537813587463298463541, 10.67695269329044250115577494476, 11.86940316563732175546712129602

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