Properties

Label 2-300-75.53-c1-0-7
Degree $2$
Conductor $300$
Sign $0.610 + 0.791i$
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  + (1.29 − 1.15i)3-s + (1.99 − 1.00i)5-s + (0.814 − 0.814i)7-s + (0.345 − 2.98i)9-s + (−3.46 − 1.12i)11-s + (−2.63 + 5.16i)13-s + (1.42 − 3.60i)15-s + (0.902 + 0.142i)17-s + (4.29 + 5.91i)19-s + (0.115 − 1.99i)21-s + (−3.05 − 6.00i)23-s + (2.97 − 4.02i)25-s + (−2.98 − 4.25i)27-s + (4.30 + 3.12i)29-s + (−1.78 + 1.29i)31-s + ⋯
L(s)  = 1  + (0.746 − 0.665i)3-s + (0.892 − 0.450i)5-s + (0.307 − 0.307i)7-s + (0.115 − 0.993i)9-s + (−1.04 − 0.339i)11-s + (−0.730 + 1.43i)13-s + (0.367 − 0.930i)15-s + (0.218 + 0.0346i)17-s + (0.986 + 1.35i)19-s + (0.0251 − 0.434i)21-s + (−0.637 − 1.25i)23-s + (0.594 − 0.804i)25-s + (−0.574 − 0.818i)27-s + (0.799 + 0.580i)29-s + (−0.321 + 0.233i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 + 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.610 + 0.791i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.610 + 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61040 - 0.791799i\)
\(L(\frac12)\) \(\approx\) \(1.61040 - 0.791799i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 + 1.15i)T \)
5 \( 1 + (-1.99 + 1.00i)T \)
good7 \( 1 + (-0.814 + 0.814i)T - 7iT^{2} \)
11 \( 1 + (3.46 + 1.12i)T + (8.89 + 6.46i)T^{2} \)
13 \( 1 + (2.63 - 5.16i)T + (-7.64 - 10.5i)T^{2} \)
17 \( 1 + (-0.902 - 0.142i)T + (16.1 + 5.25i)T^{2} \)
19 \( 1 + (-4.29 - 5.91i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (3.05 + 6.00i)T + (-13.5 + 18.6i)T^{2} \)
29 \( 1 + (-4.30 - 3.12i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.78 - 1.29i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.74 - 1.39i)T + (21.7 + 29.9i)T^{2} \)
41 \( 1 + (2.66 - 0.866i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 + (3.01 + 3.01i)T + 43iT^{2} \)
47 \( 1 + (-0.998 - 6.30i)T + (-44.6 + 14.5i)T^{2} \)
53 \( 1 + (-8.04 + 1.27i)T + (50.4 - 16.3i)T^{2} \)
59 \( 1 + (1.98 + 6.10i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (4.21 - 12.9i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.976 + 6.16i)T + (-63.7 - 20.7i)T^{2} \)
71 \( 1 + (5.88 - 8.09i)T + (-21.9 - 67.5i)T^{2} \)
73 \( 1 + (8.26 - 4.21i)T + (42.9 - 59.0i)T^{2} \)
79 \( 1 + (-4.29 + 5.90i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-0.190 + 1.20i)T + (-78.9 - 25.6i)T^{2} \)
89 \( 1 + (3.41 - 10.5i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-3.84 + 0.609i)T + (92.2 - 29.9i)T^{2} \)
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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

−11.94880590084964998528066811117, −10.42683455373756335453561655475, −9.674310201649060481568031054015, −8.681803176430287014223095680449, −7.85594764835072732732184464056, −6.81358693921335150635676205150, −5.69993945477484310402609003345, −4.41095990868933869165892410779, −2.74278767862221416861319339852, −1.53890194684106736944667926933, 2.33381755058840883347216236826, 3.15620262136758452081062859435, 4.99337608660302912415467850116, 5.55151416426510452640395397785, 7.31139741728705109419256886449, 8.026884190770836470668752259068, 9.273304139974456210098885016488, 10.01355291581031987485766608930, 10.57613039972601654545209842508, 11.77095907639718033334410680701

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