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Properties

Label 2-300-3.2-c8-0-10
Degree $2$
Conductor $300$
Sign $-0.979 + 0.201i$
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  + (−79.3 + 16.2i)3-s − 3.40e3·7-s + (6.03e3 − 2.58e3i)9-s + 2.45e4i·11-s − 2.51e4·13-s + 1.27e5i·17-s + 1.10e5·19-s + (2.70e5 − 5.54e4i)21-s − 2.00e5i·23-s + (−4.36e5 + 3.03e5i)27-s + 1.85e5i·29-s + 8.00e5·31-s + (−4.00e5 − 1.94e6i)33-s + 8.59e5·37-s + (1.99e6 − 4.10e5i)39-s + ⋯
L(s)  = 1  + (−0.979 + 0.201i)3-s − 1.41·7-s + (0.919 − 0.394i)9-s + 1.67i·11-s − 0.881·13-s + 1.52i·17-s + 0.849·19-s + (1.38 − 0.285i)21-s − 0.715i·23-s + (−0.821 + 0.570i)27-s + 0.262i·29-s + 0.867·31-s + (−0.337 − 1.64i)33-s + 0.458·37-s + (0.863 − 0.177i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.979 + 0.201i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.979 + 0.201i$
Analytic conductor: \(122.213\)
Root analytic conductor: \(11.0550\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :4),\ -0.979 + 0.201i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(0.6772762991\)
\(L(\frac12)\) \(\approx\) \(0.6772762991\)
\(L(5)\) 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 \( 1 + (79.3 - 16.2i)T \)
5 \( 1 \)
good7 \( 1 + 3.40e3T + 5.76e6T^{2} \)
11 \( 1 - 2.45e4iT - 2.14e8T^{2} \)
13 \( 1 + 2.51e4T + 8.15e8T^{2} \)
17 \( 1 - 1.27e5iT - 6.97e9T^{2} \)
19 \( 1 - 1.10e5T + 1.69e10T^{2} \)
23 \( 1 + 2.00e5iT - 7.83e10T^{2} \)
29 \( 1 - 1.85e5iT - 5.00e11T^{2} \)
31 \( 1 - 8.00e5T + 8.52e11T^{2} \)
37 \( 1 - 8.59e5T + 3.51e12T^{2} \)
41 \( 1 - 5.07e6iT - 7.98e12T^{2} \)
43 \( 1 - 4.48e6T + 1.16e13T^{2} \)
47 \( 1 - 6.94e6iT - 2.38e13T^{2} \)
53 \( 1 + 3.64e6iT - 6.22e13T^{2} \)
59 \( 1 - 6.92e6iT - 1.46e14T^{2} \)
61 \( 1 + 2.58e6T + 1.91e14T^{2} \)
67 \( 1 - 6.22e6T + 4.06e14T^{2} \)
71 \( 1 - 3.07e7iT - 6.45e14T^{2} \)
73 \( 1 - 1.15e7T + 8.06e14T^{2} \)
79 \( 1 - 6.00e7T + 1.51e15T^{2} \)
83 \( 1 - 4.00e7iT - 2.25e15T^{2} \)
89 \( 1 + 4.48e7iT - 3.93e15T^{2} \)
97 \( 1 + 6.54e7T + 7.83e15T^{2} \)
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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

−10.65127429111601372214664981369, −9.864469097181835479251036783460, −9.479049092151255755674693612274, −7.73499242481459173526565019177, −6.76620862342048734393879870372, −6.12179183219465900455430919707, −4.86955142285351418639722898512, −4.00782640628638271963294065571, −2.59076925307346730794097385399, −1.13977255189070218217331156523, 0.25928324358819280471481949548, 0.73021278735467227023590392100, 2.62690985009728011656894349196, 3.62776114640845819478509284552, 5.13920666564151118230069926700, 5.89098586780018145148946207503, 6.81558539469759136887153839106, 7.64347157271370933112743034299, 9.185353120106507354892039587552, 9.842432334215048080102547579025

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