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Properties

Label 2-300-75.17-c1-0-4
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: $\chi_{300} (17, \cdot )$
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

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

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