Properties

Label 2-300-4.3-c4-0-35
Degree $2$
Conductor $300$
Sign $0.433 - 0.901i$
Analytic cond. $31.0109$
Root an. cond. $5.56875$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (3.89 + 0.889i)2-s − 5.19i·3-s + (14.4 + 6.93i)4-s + (4.61 − 20.2i)6-s + 44.0i·7-s + (50.0 + 39.8i)8-s − 27·9-s + 81.7i·11-s + (36.0 − 74.9i)12-s + 192.·13-s + (−39.2 + 171. i)14-s + (159. + 199. i)16-s − 415.·17-s + (−105. − 24.0i)18-s + 23.6i·19-s + ⋯
L(s)  = 1  + (0.974 + 0.222i)2-s − 0.577i·3-s + (0.901 + 0.433i)4-s + (0.128 − 0.562i)6-s + 0.899i·7-s + (0.782 + 0.622i)8-s − 0.333·9-s + 0.675i·11-s + (0.250 − 0.520i)12-s + 1.13·13-s + (−0.200 + 0.877i)14-s + (0.624 + 0.781i)16-s − 1.43·17-s + (−0.324 − 0.0740i)18-s + 0.0655i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.433 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.433 - 0.901i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.433 - 0.901i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(3.648969379\)
\(L(\frac12)\) \(\approx\) \(3.648969379\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-3.89 - 0.889i)T \)
3 \( 1 + 5.19iT \)
5 \( 1 \)
good7 \( 1 - 44.0iT - 2.40e3T^{2} \)
11 \( 1 - 81.7iT - 1.46e4T^{2} \)
13 \( 1 - 192.T + 2.85e4T^{2} \)
17 \( 1 + 415.T + 8.35e4T^{2} \)
19 \( 1 - 23.6iT - 1.30e5T^{2} \)
23 \( 1 - 525. iT - 2.79e5T^{2} \)
29 \( 1 - 254.T + 7.07e5T^{2} \)
31 \( 1 - 1.30e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.22e3T + 1.87e6T^{2} \)
41 \( 1 - 1.95e3T + 2.82e6T^{2} \)
43 \( 1 + 2.49e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.00e3iT - 4.87e6T^{2} \)
53 \( 1 + 20.8T + 7.89e6T^{2} \)
59 \( 1 + 2.21e3iT - 1.21e7T^{2} \)
61 \( 1 - 2.19e3T + 1.38e7T^{2} \)
67 \( 1 + 6.56e3iT - 2.01e7T^{2} \)
71 \( 1 - 8.34e3iT - 2.54e7T^{2} \)
73 \( 1 + 6.09e3T + 2.83e7T^{2} \)
79 \( 1 - 6.12e3iT - 3.89e7T^{2} \)
83 \( 1 + 9.94e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.74e3T + 6.27e7T^{2} \)
97 \( 1 - 1.53e3T + 8.85e7T^{2} \)
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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

−11.51710880193121555491397555814, −10.75054258017905472061227266404, −9.148496330198293848214374896116, −8.250613284245050757974298162094, −7.11068404164175609580447901842, −6.27198329275584406268819422408, −5.37118008107589207443799153961, −4.13231886079599646834636194561, −2.75359643770861887820987530872, −1.66025830407389607413596008806, 0.817387949074730910180947025335, 2.58326563858092115783505099010, 3.90239058342876020470312844531, 4.47540096983516434254482755678, 5.89916250436419924270084899074, 6.65288353636694998227398458080, 7.986103182308381943046326960365, 9.171682302932545727504133733504, 10.43381193317116185027905329375, 10.97685271648794844656290912892

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