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Properties

Label 2-300-5.2-c8-0-6
Degree $2$
Conductor $300$
Sign $0.973 - 0.229i$
Analytic cond. $122.213$
Root an. cond. $11.0550$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (33.0 − 33.0i)3-s + (−945. − 945. i)7-s − 2.18e3i·9-s − 2.69e4·11-s + (−2.78e4 + 2.78e4i)13-s + (−8.14e4 − 8.14e4i)17-s − 6.29e4i·19-s − 6.25e4·21-s + (−5.46e3 + 5.46e3i)23-s + (−7.23e4 − 7.23e4i)27-s + 1.31e6i·29-s + 1.41e6·31-s + (−8.91e5 + 8.91e5i)33-s + (−1.21e6 − 1.21e6i)37-s + 1.84e6i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (−0.393 − 0.393i)7-s − 0.333i·9-s − 1.84·11-s + (−0.976 + 0.976i)13-s + (−0.975 − 0.975i)17-s − 0.483i·19-s − 0.321·21-s + (−0.0195 + 0.0195i)23-s + (−0.136 − 0.136i)27-s + 1.86i·29-s + 1.53·31-s + (−0.751 + 0.751i)33-s + (−0.647 − 0.647i)37-s + 0.797i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.973 - 0.229i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ 0.973 - 0.229i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(1.208855288\)
\(L(\frac12)\) \(\approx\) \(1.208855288\)
\(L(5)\) 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 \( 1 + (-33.0 + 33.0i)T \)
5 \( 1 \)
good7 \( 1 + (945. + 945. i)T + 5.76e6iT^{2} \)
11 \( 1 + 2.69e4T + 2.14e8T^{2} \)
13 \( 1 + (2.78e4 - 2.78e4i)T - 8.15e8iT^{2} \)
17 \( 1 + (8.14e4 + 8.14e4i)T + 6.97e9iT^{2} \)
19 \( 1 + 6.29e4iT - 1.69e10T^{2} \)
23 \( 1 + (5.46e3 - 5.46e3i)T - 7.83e10iT^{2} \)
29 \( 1 - 1.31e6iT - 5.00e11T^{2} \)
31 \( 1 - 1.41e6T + 8.52e11T^{2} \)
37 \( 1 + (1.21e6 + 1.21e6i)T + 3.51e12iT^{2} \)
41 \( 1 - 4.79e6T + 7.98e12T^{2} \)
43 \( 1 + (-1.55e6 + 1.55e6i)T - 1.16e13iT^{2} \)
47 \( 1 + (9.16e5 + 9.16e5i)T + 2.38e13iT^{2} \)
53 \( 1 + (1.86e6 - 1.86e6i)T - 6.22e13iT^{2} \)
59 \( 1 - 4.95e6iT - 1.46e14T^{2} \)
61 \( 1 - 1.27e7T + 1.91e14T^{2} \)
67 \( 1 + (-2.65e7 - 2.65e7i)T + 4.06e14iT^{2} \)
71 \( 1 + 1.18e7T + 6.45e14T^{2} \)
73 \( 1 + (1.23e7 - 1.23e7i)T - 8.06e14iT^{2} \)
79 \( 1 - 2.07e7iT - 1.51e15T^{2} \)
83 \( 1 + (-3.01e7 + 3.01e7i)T - 2.25e15iT^{2} \)
89 \( 1 + 6.61e7iT - 3.93e15T^{2} \)
97 \( 1 + (8.68e7 + 8.68e7i)T + 7.83e15iT^{2} \)
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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

−10.32756628875215288260505113071, −9.405047874499568772567753148492, −8.479744756676120810990864871520, −7.29374871977662245502153146751, −6.89526272924903560738091153598, −5.34006622299906063343513017342, −4.41663872926301245928266156229, −2.87810490608151287841269079179, −2.24604974731671294957245119520, −0.60688380413803788586281932634, 0.36119087161379950836120067639, 2.31862902124092940037660954538, 2.86527744528353368852981427266, 4.30760263903722854587469995026, 5.30434927496103123159414494140, 6.28134504584493088337855815866, 7.86091471470987986018381645327, 8.157183266970231233766681801849, 9.559968036563428428501416383373, 10.21337473354097192749019006255

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