Properties

Label 2-300-20.7-c1-0-9
Degree $2$
Conductor $300$
Sign $0.994 - 0.102i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.0912 + 1.41i)2-s + (−0.707 − 0.707i)3-s + (−1.98 − 0.257i)4-s + (1.06 − 0.933i)6-s + (1.86 − 1.86i)7-s + (0.544 − 2.77i)8-s + 1.00i·9-s − 0.728i·11-s + (1.22 + 1.58i)12-s + (3.12 − 3.12i)13-s + (2.46 + 2.80i)14-s + (3.86 + 1.02i)16-s + (−1.12 − 1.12i)17-s + (−1.41 − 0.0912i)18-s + 3.73·19-s + ⋯
L(s)  = 1  + (−0.0645 + 0.997i)2-s + (−0.408 − 0.408i)3-s + (−0.991 − 0.128i)4-s + (0.433 − 0.381i)6-s + (0.705 − 0.705i)7-s + (0.192 − 0.981i)8-s + 0.333i·9-s − 0.219i·11-s + (0.352 + 0.457i)12-s + (0.866 − 0.866i)13-s + (0.658 + 0.749i)14-s + (0.966 + 0.255i)16-s + (−0.272 − 0.272i)17-s + (−0.332 − 0.0215i)18-s + 0.856·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.102i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.994 - 0.102i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.994 - 0.102i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07343 + 0.0551590i\)
\(L(\frac12)\) \(\approx\) \(1.07343 + 0.0551590i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0912 - 1.41i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 \)
good7 \( 1 + (-1.86 + 1.86i)T - 7iT^{2} \)
11 \( 1 + 0.728iT - 11T^{2} \)
13 \( 1 + (-3.12 + 3.12i)T - 13iT^{2} \)
17 \( 1 + (1.12 + 1.12i)T + 17iT^{2} \)
19 \( 1 - 3.73T + 19T^{2} \)
23 \( 1 + (-5.83 - 5.83i)T + 23iT^{2} \)
29 \( 1 + 2.64iT - 29T^{2} \)
31 \( 1 + 6.01iT - 31T^{2} \)
37 \( 1 + (3.12 + 3.12i)T + 37iT^{2} \)
41 \( 1 + 4.24T + 41T^{2} \)
43 \( 1 + (5.10 + 5.10i)T + 43iT^{2} \)
47 \( 1 + (2.09 - 2.09i)T - 47iT^{2} \)
53 \( 1 + (0.484 - 0.484i)T - 53iT^{2} \)
59 \( 1 - 4.92T + 59T^{2} \)
61 \( 1 - 2.31T + 61T^{2} \)
67 \( 1 + (-5.10 + 5.10i)T - 67iT^{2} \)
71 \( 1 - 13.1iT - 71T^{2} \)
73 \( 1 + (3.96 - 3.96i)T - 73iT^{2} \)
79 \( 1 + 7.11T + 79T^{2} \)
83 \( 1 + (3.55 + 3.55i)T + 83iT^{2} \)
89 \( 1 + 1.03iT - 89T^{2} \)
97 \( 1 + (-12.5 - 12.5i)T + 97iT^{2} \)
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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

−11.66320359684530215653439218144, −10.86004132267269294669785148372, −9.787104444262962092525543273448, −8.603540112636733830065780931293, −7.71477950692649182000023515189, −7.02802658580318381656248830191, −5.80190763743849449574041827384, −5.01559884467126158865370665600, −3.63456712892434686927069440608, −1.02625442251022805317732936383, 1.58987505401084893253416031023, 3.19485994822676857198822705381, 4.55493245404450818296464370575, 5.31051580732662035429632687205, 6.74697412839849587978367286484, 8.443296103579137786047732580883, 8.929808650818418563206185918589, 10.04494354314135184464727488079, 10.97408068177144080020065719727, 11.57997187946327920731164566975

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