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Properties

Label 2-300-5.3-c10-0-23
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 + (−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

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

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

\(L(\frac{11}{2})\) \(\approx\) \(0.2537839027\)
\(L(\frac12)\) \(\approx\) \(0.2537839027\)
\(L(6)\) 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

−9.307406431166703501297868250667, −8.913566902669835027228310327159, −7.47735609903907148299462280892, −6.64988533272488173373739596527, −5.86859558407219237475922985495, −4.70651523449679355555334009511, −3.55500079927182177372132713886, −2.29249519662798119926027714755, −1.24444609141497052646315171457, −0.05761272290755584563131266527, 0.958677837887395316976400200072, 2.30078398518261263372897537297, 3.70258297903303605391579473631, 4.39507419629160923368249540941, 5.57017528326479069702734003959, 6.70613866859524465043778217562, 7.25820691153165589957600288843, 8.990890555236953952260727896976, 9.304016299573500993496416859479, 10.52157835183577028316668894833

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