Properties

Label 2-300-300.287-c2-0-71
Degree $2$
Conductor $300$
Sign $0.955 - 0.294i$
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  + (−0.553 + 1.92i)2-s + (2.45 + 1.73i)3-s + (−3.38 − 2.12i)4-s + (−4.77 − 1.47i)5-s + (−4.68 + 3.75i)6-s + (8.95 − 8.95i)7-s + (5.96 − 5.33i)8-s + (3.00 + 8.48i)9-s + (5.47 − 8.36i)10-s + (−3.41 − 2.48i)11-s + (−4.61 − 11.0i)12-s + (3.36 − 21.2i)13-s + (12.2 + 22.1i)14-s + (−9.15 − 11.8i)15-s + (6.95 + 14.4i)16-s + (0.840 + 1.64i)17-s + ⋯
L(s)  = 1  + (−0.276 + 0.960i)2-s + (0.816 + 0.576i)3-s + (−0.846 − 0.531i)4-s + (−0.955 − 0.295i)5-s + (−0.780 + 0.625i)6-s + (1.27 − 1.27i)7-s + (0.745 − 0.666i)8-s + (0.334 + 0.942i)9-s + (0.547 − 0.836i)10-s + (−0.310 − 0.225i)11-s + (−0.384 − 0.922i)12-s + (0.258 − 1.63i)13-s + (0.875 + 1.58i)14-s + (−0.610 − 0.792i)15-s + (0.434 + 0.900i)16-s + (0.0494 + 0.0970i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.955 - 0.294i$
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: \((2,\ 300,\ (\ :1),\ 0.955 - 0.294i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.57956 + 0.237576i\)
\(L(\frac12)\) \(\approx\) \(1.57956 + 0.237576i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.553 - 1.92i)T \)
3 \( 1 + (-2.45 - 1.73i)T \)
5 \( 1 + (4.77 + 1.47i)T \)
good7 \( 1 + (-8.95 + 8.95i)T - 49iT^{2} \)
11 \( 1 + (3.41 + 2.48i)T + (37.3 + 115. i)T^{2} \)
13 \( 1 + (-3.36 + 21.2i)T + (-160. - 52.2i)T^{2} \)
17 \( 1 + (-0.840 - 1.64i)T + (-169. + 233. i)T^{2} \)
19 \( 1 + (-0.691 + 2.12i)T + (-292. - 212. i)T^{2} \)
23 \( 1 + (-32.6 + 5.16i)T + (503. - 163. i)T^{2} \)
29 \( 1 + (5.26 + 16.2i)T + (-680. + 494. i)T^{2} \)
31 \( 1 + (29.3 + 9.52i)T + (777. + 564. i)T^{2} \)
37 \( 1 + (-16.3 - 2.58i)T + (1.30e3 + 423. i)T^{2} \)
41 \( 1 + (-23.0 - 31.6i)T + (-519. + 1.59e3i)T^{2} \)
43 \( 1 + (9.14 + 9.14i)T + 1.84e3iT^{2} \)
47 \( 1 + (29.2 - 57.4i)T + (-1.29e3 - 1.78e3i)T^{2} \)
53 \( 1 + (-22.1 + 43.4i)T + (-1.65e3 - 2.27e3i)T^{2} \)
59 \( 1 + (22.5 + 30.9i)T + (-1.07e3 + 3.31e3i)T^{2} \)
61 \( 1 + (-26.1 - 18.9i)T + (1.14e3 + 3.53e3i)T^{2} \)
67 \( 1 + (-4.85 - 9.53i)T + (-2.63e3 + 3.63e3i)T^{2} \)
71 \( 1 + (13.5 + 41.6i)T + (-4.07e3 + 2.96e3i)T^{2} \)
73 \( 1 + (-114. + 18.1i)T + (5.06e3 - 1.64e3i)T^{2} \)
79 \( 1 + (-6.99 - 21.5i)T + (-5.04e3 + 3.66e3i)T^{2} \)
83 \( 1 + (25.1 + 49.3i)T + (-4.04e3 + 5.57e3i)T^{2} \)
89 \( 1 + (-108. - 79.1i)T + (2.44e3 + 7.53e3i)T^{2} \)
97 \( 1 + (41.1 - 80.7i)T + (-5.53e3 - 7.61e3i)T^{2} \)
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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

−10.99081950838273112175448087474, −10.71137678315792578746880122357, −9.460582951548744515442188717644, −8.177585337335094949688565621550, −8.026189165576172363370681896724, −7.19766812715957490069968350847, −5.26047875755273477893980678303, −4.52524528193130183462054998986, −3.49579372122681568411514490624, −0.893511011922179994038959959174, 1.58499660048394213351140023325, 2.61737667817859674097889964939, 3.91017428210182147955305693393, 5.06244962376188228063863627088, 7.02836039549527130249571382664, 7.917675157680249350179246995283, 8.803900466477558589230267384851, 9.201677662095175008270112842289, 10.87955250046074908403922546360, 11.60091436368376232489734540613

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