Properties

Label 2-300-300.287-c2-0-95
Degree $2$
Conductor $300$
Sign $0.341 + 0.939i$
Analytic cond. $8.17440$
Root an. cond. $2.85909$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.94 − 0.462i)2-s + (0.146 − 2.99i)3-s + (3.57 − 1.79i)4-s + (2.39 + 4.38i)5-s + (−1.10 − 5.89i)6-s + (4.78 − 4.78i)7-s + (6.12 − 5.15i)8-s + (−8.95 − 0.875i)9-s + (6.68 + 7.43i)10-s + (0.198 + 0.143i)11-s + (−4.86 − 10.9i)12-s + (0.392 − 2.47i)13-s + (7.10 − 11.5i)14-s + (13.5 − 6.53i)15-s + (9.52 − 12.8i)16-s + (3.53 + 6.93i)17-s + ⋯
L(s)  = 1  + (0.972 − 0.231i)2-s + (0.0486 − 0.998i)3-s + (0.893 − 0.449i)4-s + (0.478 + 0.877i)5-s + (−0.183 − 0.983i)6-s + (0.684 − 0.684i)7-s + (0.765 − 0.643i)8-s + (−0.995 − 0.0972i)9-s + (0.668 + 0.743i)10-s + (0.0180 + 0.0130i)11-s + (−0.405 − 0.914i)12-s + (0.0301 − 0.190i)13-s + (0.507 − 0.823i)14-s + (0.900 − 0.435i)15-s + (0.595 − 0.803i)16-s + (0.207 + 0.407i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.341 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.341 + 0.939i$
Analytic conductor: \(8.17440\)
Root analytic conductor: \(2.85909\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (287, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1),\ 0.341 + 0.939i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.74206 - 1.92100i\)
\(L(\frac12)\) \(\approx\) \(2.74206 - 1.92100i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.94 + 0.462i)T \)
3 \( 1 + (-0.146 + 2.99i)T \)
5 \( 1 + (-2.39 - 4.38i)T \)
good7 \( 1 + (-4.78 + 4.78i)T - 49iT^{2} \)
11 \( 1 + (-0.198 - 0.143i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (-0.392 + 2.47i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (-3.53 - 6.93i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (2.59 - 7.98i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-4.23 + 0.670i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (8.46 + 26.0i)T + (-680. + 494. i)T^{2} \)
31 \( 1 + (30.0 + 9.76i)T + (777. + 564. i)T^{2} \)
37 \( 1 + (-42.6 - 6.76i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (-3.16 - 4.36i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + (54.3 + 54.3i)T + 1.84e3iT^{2} \)
47 \( 1 + (31.7 - 62.3i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (32.1 - 63.0i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (-53.1 - 73.1i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (2.05 + 1.49i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (-39.7 - 77.9i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (6.51 + 20.0i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-94.4 + 14.9i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (-39.6 - 121. i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (-34.9 - 68.5i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (43.8 + 31.8i)T + (2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (-7.39 + 14.5i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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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

−11.33206952811605750786299621034, −10.85291408155620766173296310538, −9.760813345678261971068134280765, −8.008215505737220438664525200248, −7.26380874271967649660307950493, −6.33358983869049128026706496391, −5.48609694693803736286615229189, −3.92223391277001673046953644648, −2.62113877552800044092869633230, −1.45542451462396478463151592443, 2.07929595969481071839879240941, 3.55175948273550671827234142936, 4.98765704653340186545711736892, 5.14181760165651446769377684694, 6.41359263320345420237504661911, 8.015237398530684619735409379000, 8.838153786357561470337493200272, 9.790520433571373793904957557479, 11.07238608250031934399286016539, 11.67695459321769988159725764769

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