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Properties

Label 2-300-5.3-c10-0-25
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 + (1.75e4 − 1.75e4i)7-s + 1.96e4i·9-s − 2.45e5·11-s + (2.58e4 + 2.58e4i)13-s + (−1.01e6 + 1.01e6i)17-s + 2.08e6i·19-s + 3.48e6·21-s + (−4.16e6 − 4.16e6i)23-s + (−1.95e6 + 1.95e6i)27-s + 2.48e7i·29-s + 4.39e7·31-s + (−2.43e7 − 2.43e7i)33-s + (8.21e7 − 8.21e7i)37-s + 5.13e6i·39-s + ⋯
L(s)  = 1  + (0.408 + 0.408i)3-s + (1.04 − 1.04i)7-s + 0.333i·9-s − 1.52·11-s + (0.0697 + 0.0697i)13-s + (−0.715 + 0.715i)17-s + 0.842i·19-s + 0.854·21-s + (−0.646 − 0.646i)23-s + (−0.136 + 0.136i)27-s + 1.21i·29-s + 1.53·31-s + (−0.621 − 0.621i)33-s + (1.18 − 1.18i)37-s + 0.0569i·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

\(L(\frac{11}{2})\) \(\approx\) \(1.782276393\)
\(L(\frac12)\) \(\approx\) \(1.782276393\)
\(L(6)\) 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 \( 1 + (-1.75e4 + 1.75e4i)T - 2.82e8iT^{2} \)
11 \( 1 + 2.45e5T + 2.59e10T^{2} \)
13 \( 1 + (-2.58e4 - 2.58e4i)T + 1.37e11iT^{2} \)
17 \( 1 + (1.01e6 - 1.01e6i)T - 2.01e12iT^{2} \)
19 \( 1 - 2.08e6iT - 6.13e12T^{2} \)
23 \( 1 + (4.16e6 + 4.16e6i)T + 4.14e13iT^{2} \)
29 \( 1 - 2.48e7iT - 4.20e14T^{2} \)
31 \( 1 - 4.39e7T + 8.19e14T^{2} \)
37 \( 1 + (-8.21e7 + 8.21e7i)T - 4.80e15iT^{2} \)
41 \( 1 - 3.41e7T + 1.34e16T^{2} \)
43 \( 1 + (5.69e7 + 5.69e7i)T + 2.16e16iT^{2} \)
47 \( 1 + (-1.56e8 + 1.56e8i)T - 5.25e16iT^{2} \)
53 \( 1 + (9.10e7 + 9.10e7i)T + 1.74e17iT^{2} \)
59 \( 1 - 1.50e8iT - 5.11e17T^{2} \)
61 \( 1 - 1.58e9T + 7.13e17T^{2} \)
67 \( 1 + (9.08e7 - 9.08e7i)T - 1.82e18iT^{2} \)
71 \( 1 + 3.33e9T + 3.25e18T^{2} \)
73 \( 1 + (6.15e8 + 6.15e8i)T + 4.29e18iT^{2} \)
79 \( 1 - 1.25e9iT - 9.46e18T^{2} \)
83 \( 1 + (2.20e9 + 2.20e9i)T + 1.55e19iT^{2} \)
89 \( 1 + 5.39e9iT - 3.11e19T^{2} \)
97 \( 1 + (-6.66e9 + 6.66e9i)T - 7.37e19iT^{2} \)
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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

−10.14262266065148984456138547459, −8.589370741099321861391767135087, −8.038906103155222609109491484776, −7.17954190862511455153631393547, −5.75310096412262881617470159276, −4.65826921002210488642947566201, −3.98873393088773463962385771549, −2.64167705447739160821969742663, −1.61846909695636509824643086209, −0.32010713503366265159692249853, 0.977023754214709853436064580166, 2.38458740473131869517934322496, 2.65903231012188043093963934027, 4.48813014340547680138975122871, 5.29445354303400184376424633843, 6.35099031836920658792894541110, 7.72699966815476747412304162580, 8.155964706854329914234688855879, 9.129231307281440071153523179915, 10.15655797777156196533320549315

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