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Properties

Label 2-300-5.2-c10-0-11
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 + (7.20e3 + 7.20e3i)7-s − 1.96e4i·9-s + 2.10e5·11-s + (1.50e5 − 1.50e5i)13-s + (1.85e6 + 1.85e6i)17-s + 3.28e6i·19-s + 1.42e6·21-s + (−6.83e6 + 6.83e6i)23-s + (−1.95e6 − 1.95e6i)27-s + 5.70e6i·29-s − 5.12e7·31-s + (2.08e7 − 2.08e7i)33-s + (6.63e7 + 6.63e7i)37-s − 2.97e7i·39-s + ⋯
L(s)  = 1  + (0.408 − 0.408i)3-s + (0.428 + 0.428i)7-s − 0.333i·9-s + 1.30·11-s + (0.404 − 0.404i)13-s + (1.30 + 1.30i)17-s + 1.32i·19-s + 0.349·21-s + (−1.06 + 1.06i)23-s + (−0.136 − 0.136i)27-s + 0.278i·29-s − 1.79·31-s + (0.533 − 0.533i)33-s + (0.957 + 0.957i)37-s − 0.330i·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: $\chi_{300} (157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :5),\ 0.130 - 0.991i)\)

Particular Values

\(L(\frac{11}{2})\) \(\approx\) \(2.692740556\)
\(L(\frac12)\) \(\approx\) \(2.692740556\)
\(L(6)\) 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 \( 1 + (-7.20e3 - 7.20e3i)T + 2.82e8iT^{2} \)
11 \( 1 - 2.10e5T + 2.59e10T^{2} \)
13 \( 1 + (-1.50e5 + 1.50e5i)T - 1.37e11iT^{2} \)
17 \( 1 + (-1.85e6 - 1.85e6i)T + 2.01e12iT^{2} \)
19 \( 1 - 3.28e6iT - 6.13e12T^{2} \)
23 \( 1 + (6.83e6 - 6.83e6i)T - 4.14e13iT^{2} \)
29 \( 1 - 5.70e6iT - 4.20e14T^{2} \)
31 \( 1 + 5.12e7T + 8.19e14T^{2} \)
37 \( 1 + (-6.63e7 - 6.63e7i)T + 4.80e15iT^{2} \)
41 \( 1 - 1.97e8T + 1.34e16T^{2} \)
43 \( 1 + (8.01e7 - 8.01e7i)T - 2.16e16iT^{2} \)
47 \( 1 + (1.70e8 + 1.70e8i)T + 5.25e16iT^{2} \)
53 \( 1 + (1.87e8 - 1.87e8i)T - 1.74e17iT^{2} \)
59 \( 1 - 3.54e8iT - 5.11e17T^{2} \)
61 \( 1 + 1.18e9T + 7.13e17T^{2} \)
67 \( 1 + (1.62e9 + 1.62e9i)T + 1.82e18iT^{2} \)
71 \( 1 - 3.16e8T + 3.25e18T^{2} \)
73 \( 1 + (-5.47e8 + 5.47e8i)T - 4.29e18iT^{2} \)
79 \( 1 - 4.50e9iT - 9.46e18T^{2} \)
83 \( 1 + (2.12e9 - 2.12e9i)T - 1.55e19iT^{2} \)
89 \( 1 + 5.13e9iT - 3.11e19T^{2} \)
97 \( 1 + (6.06e9 + 6.06e9i)T + 7.37e19iT^{2} \)
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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

−10.08695929143636539857488592246, −9.227378662740369072788778418161, −8.186227673536567015670777354877, −7.65266694353813455707890337411, −6.21086336925248344760164617306, −5.65668517438570318542852534172, −3.99103325104115143826452071079, −3.34310132452764679466233626074, −1.68870756677645734754313475189, −1.34454216797751286127406579960, 0.44605562088508565210871805378, 1.50384031707441918479476374739, 2.75597772226331603044671171404, 3.91081068442885910244015815384, 4.62303504340886623906910174372, 5.89513662548288873286971996694, 7.05792985237545719348886745981, 7.88032019237678287882884058541, 9.132945568385846195737739325136, 9.500455500411450133457419319332

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