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Properties

Label 2-300-15.2-c7-0-27
Degree $2$
Conductor $300$
Sign $0.999 + 0.0327i$
Analytic cond. $93.7155$
Root an. cond. $9.68067$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (23.2 + 40.5i)3-s + (208. + 208. i)7-s + (−1.10e3 + 1.88e3i)9-s − 7.00e3i·11-s + (−9.87e3 + 9.87e3i)13-s + (1.75e4 − 1.75e4i)17-s − 2.43e4i·19-s + (−3.59e3 + 1.32e4i)21-s + (−1.57e4 − 1.57e4i)23-s + (−1.02e5 − 854. i)27-s + 1.22e5·29-s + 2.28e5·31-s + (2.84e5 − 1.62e5i)33-s + (1.55e5 + 1.55e5i)37-s + (−6.30e5 − 1.70e5i)39-s + ⋯
L(s)  = 1  + (0.497 + 0.867i)3-s + (0.229 + 0.229i)7-s + (−0.504 + 0.863i)9-s − 1.58i·11-s + (−1.24 + 1.24i)13-s + (0.864 − 0.864i)17-s − 0.815i·19-s + (−0.0848 + 0.313i)21-s + (−0.270 − 0.270i)23-s + (−0.999 − 0.00835i)27-s + 0.932·29-s + 1.37·31-s + (1.37 − 0.789i)33-s + (0.503 + 0.503i)37-s + (−1.70 − 0.460i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0327i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0327i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.999 + 0.0327i$
Analytic conductor: \(93.7155\)
Root analytic conductor: \(9.68067\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :7/2),\ 0.999 + 0.0327i)\)

Particular Values

\(L(4)\) \(\approx\) \(2.311953137\)
\(L(\frac12)\) \(\approx\) \(2.311953137\)
\(L(\frac{9}{2})\) 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 \( 1 + (-23.2 - 40.5i)T \)
5 \( 1 \)
good7 \( 1 + (-208. - 208. i)T + 8.23e5iT^{2} \)
11 \( 1 + 7.00e3iT - 1.94e7T^{2} \)
13 \( 1 + (9.87e3 - 9.87e3i)T - 6.27e7iT^{2} \)
17 \( 1 + (-1.75e4 + 1.75e4i)T - 4.10e8iT^{2} \)
19 \( 1 + 2.43e4iT - 8.93e8T^{2} \)
23 \( 1 + (1.57e4 + 1.57e4i)T + 3.40e9iT^{2} \)
29 \( 1 - 1.22e5T + 1.72e10T^{2} \)
31 \( 1 - 2.28e5T + 2.75e10T^{2} \)
37 \( 1 + (-1.55e5 - 1.55e5i)T + 9.49e10iT^{2} \)
41 \( 1 + 7.25e5iT - 1.94e11T^{2} \)
43 \( 1 + (4.30e5 - 4.30e5i)T - 2.71e11iT^{2} \)
47 \( 1 + (2.43e4 - 2.43e4i)T - 5.06e11iT^{2} \)
53 \( 1 + (-4.64e5 - 4.64e5i)T + 1.17e12iT^{2} \)
59 \( 1 + 7.80e4T + 2.48e12T^{2} \)
61 \( 1 - 1.98e6T + 3.14e12T^{2} \)
67 \( 1 + (2.14e6 + 2.14e6i)T + 6.06e12iT^{2} \)
71 \( 1 + 8.70e5iT - 9.09e12T^{2} \)
73 \( 1 + (-3.29e6 + 3.29e6i)T - 1.10e13iT^{2} \)
79 \( 1 + 4.50e5iT - 1.92e13T^{2} \)
83 \( 1 + (-4.28e6 - 4.28e6i)T + 2.71e13iT^{2} \)
89 \( 1 + 5.27e6T + 4.42e13T^{2} \)
97 \( 1 + (-4.71e5 - 4.71e5i)T + 8.07e13iT^{2} \)
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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

−10.38873510278978244080450147278, −9.520150454662476727381786111333, −8.746797029986798372571295586970, −7.890442063782921837658653430209, −6.60552722019064203052270309833, −5.28724687442886745827668763042, −4.51468192467794802749747868703, −3.21430078196407803741989300320, −2.35963418434387050130189847225, −0.56910825273293113330278835741, 0.926755043266091466508941395935, 2.00865529148380358338303094750, 3.05670242728826814524836633897, 4.44357998729855567999308416033, 5.67238421776090120548509005251, 6.87240463179004998676653523501, 7.75488344024152405549965385882, 8.239940733106057334012492099863, 9.854941124868135468378537906960, 10.14034100223653631208423224497

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