Properties

Label 2-300-5.4-c3-0-2
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 + 32i·7-s − 9·9-s + 36·11-s + 10i·13-s − 78i·17-s − 140·19-s − 96·21-s + 192i·23-s − 27i·27-s − 6·29-s − 16·31-s + 108i·33-s − 34i·37-s − 30·39-s + ⋯
L(s)  = 1  + 0.577i·3-s + 1.72i·7-s − 0.333·9-s + 0.986·11-s + 0.213i·13-s − 1.11i·17-s − 1.69·19-s − 0.997·21-s + 1.74i·23-s − 0.192i·27-s − 0.0384·29-s − 0.0926·31-s + 0.569i·33-s − 0.151i·37-s − 0.123·39-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

\(L(2)\) \(\approx\) \(1.249069296\)
\(L(\frac12)\) \(\approx\) \(1.249069296\)
\(L(\frac{5}{2})\) 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 \( 1 - 32iT - 343T^{2} \)
11 \( 1 - 36T + 1.33e3T^{2} \)
13 \( 1 - 10iT - 2.19e3T^{2} \)
17 \( 1 + 78iT - 4.91e3T^{2} \)
19 \( 1 + 140T + 6.85e3T^{2} \)
23 \( 1 - 192iT - 1.21e4T^{2} \)
29 \( 1 + 6T + 2.43e4T^{2} \)
31 \( 1 + 16T + 2.97e4T^{2} \)
37 \( 1 + 34iT - 5.06e4T^{2} \)
41 \( 1 + 390T + 6.89e4T^{2} \)
43 \( 1 - 52iT - 7.95e4T^{2} \)
47 \( 1 - 408iT - 1.03e5T^{2} \)
53 \( 1 - 114iT - 1.48e5T^{2} \)
59 \( 1 + 516T + 2.05e5T^{2} \)
61 \( 1 + 58T + 2.26e5T^{2} \)
67 \( 1 + 892iT - 3.00e5T^{2} \)
71 \( 1 + 120T + 3.57e5T^{2} \)
73 \( 1 - 646iT - 3.89e5T^{2} \)
79 \( 1 - 1.16e3T + 4.93e5T^{2} \)
83 \( 1 - 732iT - 5.71e5T^{2} \)
89 \( 1 - 1.59e3T + 7.04e5T^{2} \)
97 \( 1 - 194iT - 9.12e5T^{2} \)
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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

−11.75307123070010315173154494862, −10.90725182594560341120921138472, −9.411476244455909597423906338065, −9.204288583641642825599473112711, −8.125320288630902359250080874159, −6.61932832013985628643186946535, −5.69545931301681462303955904762, −4.66716437551174447110892438396, −3.28836453152729999704062405703, −1.97244379491445672001452966247, 0.45303732462648160499282063648, 1.79576518050718373837789698804, 3.67082297408562190581801166016, 4.52385835277050934959864041572, 6.36392499259782491146598848347, 6.81047141401893679036521667879, 8.005441945174035097073673731921, 8.800520609734089177588183491482, 10.36467604784341250172957116832, 10.64920102579524409541995821930

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