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Properties

Label 2-300-5.3-c10-0-20
Degree $2$
Conductor $300$
Sign $0.130 + 0.991i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.130 + 0.991i$
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: \((2,\ 300,\ (\ :5),\ 0.130 + 0.991i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(0.09801141793\)
\(L(\frac12)\) \(\approx\) \(0.09801141793\)
\(L(6)\) 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

−9.586251845401974196986518942874, −8.947870162747149224009619753060, −8.202189641931287490551574261945, −6.69299112446698786687352438973, −6.05842302878952400694432593771, −4.78027745135774361765218469886, −3.69929546494849039652791917942, −2.72603063893841701305940344257, −1.77399157050840749169448939855, −0.02060980301419660918878602185, 0.76905404561731283638452951763, 2.08458463539124668187809122682, 3.27523491394649463478199570869, 4.00019244736142167171040106210, 5.45393732248494986490815866157, 6.65822800380000181149100431227, 7.23273469627439712364209730476, 8.299413942669108794843410613528, 9.315031216240541736581546993289, 10.15859263059275052825169931150

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