Properties

Label 2-300-5.4-c3-0-1
Degree $2$
Conductor $300$
Sign $-0.894 - 0.447i$
Analytic cond. $17.7005$
Root an. cond. $4.20720$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3i·3-s − 28i·7-s − 9·9-s − 24·11-s + 70i·13-s + 102i·17-s − 20·19-s + 84·21-s + 72i·23-s − 27i·27-s − 306·29-s − 136·31-s − 72i·33-s − 214i·37-s − 210·39-s + ⋯
L(s)  = 1  + 0.577i·3-s − 1.51i·7-s − 0.333·9-s − 0.657·11-s + 1.49i·13-s + 1.45i·17-s − 0.241·19-s + 0.872·21-s + 0.652i·23-s − 0.192i·27-s − 1.95·29-s − 0.787·31-s − 0.379i·33-s − 0.950i·37-s − 0.862·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $-0.894 - 0.447i$
Analytic conductor: \(17.7005\)
Root analytic conductor: \(4.20720\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (49, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :3/2),\ -0.894 - 0.447i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.6168154141\)
\(L(\frac12)\) \(\approx\) \(0.6168154141\)
\(L(\frac{5}{2})\) 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 \( 1 - 3iT \)
5 \( 1 \)
good7 \( 1 + 28iT - 343T^{2} \)
11 \( 1 + 24T + 1.33e3T^{2} \)
13 \( 1 - 70iT - 2.19e3T^{2} \)
17 \( 1 - 102iT - 4.91e3T^{2} \)
19 \( 1 + 20T + 6.85e3T^{2} \)
23 \( 1 - 72iT - 1.21e4T^{2} \)
29 \( 1 + 306T + 2.43e4T^{2} \)
31 \( 1 + 136T + 2.97e4T^{2} \)
37 \( 1 + 214iT - 5.06e4T^{2} \)
41 \( 1 + 150T + 6.89e4T^{2} \)
43 \( 1 - 292iT - 7.95e4T^{2} \)
47 \( 1 + 72iT - 1.03e5T^{2} \)
53 \( 1 - 414iT - 1.48e5T^{2} \)
59 \( 1 - 744T + 2.05e5T^{2} \)
61 \( 1 + 418T + 2.26e5T^{2} \)
67 \( 1 - 188iT - 3.00e5T^{2} \)
71 \( 1 - 480T + 3.57e5T^{2} \)
73 \( 1 + 434iT - 3.89e5T^{2} \)
79 \( 1 + 1.35e3T + 4.93e5T^{2} \)
83 \( 1 - 612iT - 5.71e5T^{2} \)
89 \( 1 - 30T + 7.04e5T^{2} \)
97 \( 1 + 286iT - 9.12e5T^{2} \)
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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

−11.34064297322135270274776709491, −10.79881034349280661534942936398, −9.931593722838556417466438636177, −9.014578247177705852645739610895, −7.79503360970758461673710478982, −6.94549049681046813404907980585, −5.68441340119835387532225692903, −4.29532029313424573080235283742, −3.70522203594383587188523298040, −1.73700742106261002077288777173, 0.21581774694381423481913778020, 2.19196243165657908107263875991, 3.12361369486874690740920191813, 5.21092384867302336260013302716, 5.69885639492656934443017549807, 7.07107147013258066738460917740, 8.067908766606554171168212463315, 8.870127374877108429017804567616, 9.904395766119424214776347652034, 11.08878124790016621636380164670

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