Properties

Label 2-300-300.179-c1-0-14
Degree $2$
Conductor $300$
Sign $0.871 - 0.490i$
Analytic cond. $2.39551$
Root an. cond. $1.54774$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.0273 − 1.41i)2-s + (1.04 + 1.38i)3-s + (−1.99 − 0.0772i)4-s + (−1.34 + 1.78i)5-s + (1.98 − 1.44i)6-s + 2.98·7-s + (−0.163 + 2.82i)8-s + (−0.811 + 2.88i)9-s + (2.48 + 1.95i)10-s + (−0.988 + 3.04i)11-s + (−1.98 − 2.83i)12-s + (−3.49 + 1.13i)13-s + (0.0815 − 4.21i)14-s + (−3.87 + 0.0113i)15-s + (3.98 + 0.308i)16-s + (4.35 − 3.16i)17-s + ⋯
L(s)  = 1  + (0.0193 − 0.999i)2-s + (0.603 + 0.797i)3-s + (−0.999 − 0.0386i)4-s + (−0.601 + 0.798i)5-s + (0.808 − 0.588i)6-s + 1.12·7-s + (−0.0579 + 0.998i)8-s + (−0.270 + 0.962i)9-s + (0.787 + 0.616i)10-s + (−0.298 + 0.917i)11-s + (−0.572 − 0.819i)12-s + (−0.968 + 0.314i)13-s + (0.0217 − 1.12i)14-s + (−0.999 + 0.00292i)15-s + (0.997 + 0.0771i)16-s + (1.05 − 0.766i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 - 0.490i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.871 - 0.490i$
Analytic conductor: \(2.39551\)
Root analytic conductor: \(1.54774\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :1/2),\ 0.871 - 0.490i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29958 + 0.340580i\)
\(L(\frac12)\) \(\approx\) \(1.29958 + 0.340580i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0273 + 1.41i)T \)
3 \( 1 + (-1.04 - 1.38i)T \)
5 \( 1 + (1.34 - 1.78i)T \)
good7 \( 1 - 2.98T + 7T^{2} \)
11 \( 1 + (0.988 - 3.04i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (3.49 - 1.13i)T + (10.5 - 7.64i)T^{2} \)
17 \( 1 + (-4.35 + 3.16i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.79 - 2.46i)T + (-5.87 + 18.0i)T^{2} \)
23 \( 1 + (-4.56 - 1.48i)T + (18.6 + 13.5i)T^{2} \)
29 \( 1 + (-3.22 + 4.43i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.59 + 3.57i)T + (-9.57 + 29.4i)T^{2} \)
37 \( 1 + (6.75 - 2.19i)T + (29.9 - 21.7i)T^{2} \)
41 \( 1 + (-2.93 + 0.953i)T + (33.1 - 24.0i)T^{2} \)
43 \( 1 - 9.03T + 43T^{2} \)
47 \( 1 + (2.83 - 3.90i)T + (-14.5 - 44.6i)T^{2} \)
53 \( 1 + (5.43 + 3.95i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.32 + 4.08i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-3.92 + 12.0i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (10.6 - 7.75i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-0.297 - 0.216i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (-6.69 - 2.17i)T + (59.0 + 42.9i)T^{2} \)
79 \( 1 + (-8.73 + 12.0i)T + (-24.4 - 75.1i)T^{2} \)
83 \( 1 + (-2.22 - 3.05i)T + (-25.6 + 78.9i)T^{2} \)
89 \( 1 + (-10.2 - 3.32i)T + (72.0 + 52.3i)T^{2} \)
97 \( 1 + (-0.944 + 1.30i)T + (-29.9 - 92.2i)T^{2} \)
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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

−11.63389133113531945171908448658, −10.88905425977467632837363295572, −10.00819925187345665999678998830, −9.369437419301475346809948971597, −7.978365444892803535740290737761, −7.52479547437424796251246900595, −5.18232792511467275585955701717, −4.48495453787221885372779232958, −3.27715152377244075905928400671, −2.17550944985638922250700634869, 1.03958302071976022035743981580, 3.33030033910852089591326736364, 4.80309329884803045115407286968, 5.63613021893475416324909855774, 7.17020174939546127391389592639, 7.82151983407517568565165009102, 8.509830056994183590048927827879, 9.180914271484144815369508201842, 10.74068878615606937825034808724, 12.10565563130904821991137956930

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