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Properties

Label 2-300-5.3-c10-0-27
Degree $2$
Conductor $300$
Sign $-0.945 + 0.326i$
Analytic cond. $190.607$
Root an. cond. $13.8060$
Motivic weight $10$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−99.2 − 99.2i)3-s + (1.73e4 − 1.73e4i)7-s + 1.96e4i·9-s − 2.50e4·11-s + (−2.00e5 − 2.00e5i)13-s + (1.64e6 − 1.64e6i)17-s − 1.78e6i·19-s − 3.43e6·21-s + (2.10e6 + 2.10e6i)23-s + (1.95e6 − 1.95e6i)27-s + 4.57e5i·29-s − 1.63e7·31-s + (2.48e6 + 2.48e6i)33-s + (3.60e7 − 3.60e7i)37-s + 3.98e7i·39-s + ⋯
L(s)  = 1  + (−0.408 − 0.408i)3-s + (1.03 − 1.03i)7-s + 0.333i·9-s − 0.155·11-s + (−0.540 − 0.540i)13-s + (1.15 − 1.15i)17-s − 0.719i·19-s − 0.841·21-s + (0.327 + 0.327i)23-s + (0.136 − 0.136i)27-s + 0.0223i·29-s − 0.569·31-s + (0.0634 + 0.0634i)33-s + (0.519 − 0.519i)37-s + 0.441i·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.945 + 0.326i$
Analytic conductor: \(190.607\)
Root analytic conductor: \(13.8060\)
Motivic weight: \(10\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (193, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :5),\ -0.945 + 0.326i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(1.776191594\)
\(L(\frac12)\) \(\approx\) \(1.776191594\)
\(L(6)\) 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 \( 1 + (99.2 + 99.2i)T \)
5 \( 1 \)
good7 \( 1 + (-1.73e4 + 1.73e4i)T - 2.82e8iT^{2} \)
11 \( 1 + 2.50e4T + 2.59e10T^{2} \)
13 \( 1 + (2.00e5 + 2.00e5i)T + 1.37e11iT^{2} \)
17 \( 1 + (-1.64e6 + 1.64e6i)T - 2.01e12iT^{2} \)
19 \( 1 + 1.78e6iT - 6.13e12T^{2} \)
23 \( 1 + (-2.10e6 - 2.10e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 4.57e5iT - 4.20e14T^{2} \)
31 \( 1 + 1.63e7T + 8.19e14T^{2} \)
37 \( 1 + (-3.60e7 + 3.60e7i)T - 4.80e15iT^{2} \)
41 \( 1 + 1.20e8T + 1.34e16T^{2} \)
43 \( 1 + (1.74e8 + 1.74e8i)T + 2.16e16iT^{2} \)
47 \( 1 + (-2.51e8 + 2.51e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (-1.67e8 - 1.67e8i)T + 1.74e17iT^{2} \)
59 \( 1 + 9.73e5iT - 5.11e17T^{2} \)
61 \( 1 - 7.13e8T + 7.13e17T^{2} \)
67 \( 1 + (3.85e8 - 3.85e8i)T - 1.82e18iT^{2} \)
71 \( 1 - 2.86e9T + 3.25e18T^{2} \)
73 \( 1 + (-2.05e9 - 2.05e9i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.26e8iT - 9.46e18T^{2} \)
83 \( 1 + (1.02e9 + 1.02e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 4.06e9iT - 3.11e19T^{2} \)
97 \( 1 + (1.33e9 - 1.33e9i)T - 7.37e19iT^{2} \)
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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

−9.779568155622515695578748438329, −8.444741030648171431133401556737, −7.43298586223730647084567056893, −7.07121966386355792595180045548, −5.43577891198701288866187492395, −4.90941181798241637998737210382, −3.59267984676088997062831329160, −2.27391834606179108544622752445, −1.05252320846996518449768010455, −0.38831492963380222619969679053, 1.24173785114494554175147296270, 2.18284055351329107015592592547, 3.52695488746028923543718967818, 4.73242530581545176068652869177, 5.46154450083789185662753361900, 6.37199134045208175799686180780, 7.78886443098712012229810430671, 8.499409261173675386903919825543, 9.562310253178935364847351054976, 10.43446498307240545859821214210

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