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Properties

Label 2-300-20.19-c4-0-23
Degree $2$
Conductor $300$
Sign $0.999 - 0.0153i$
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  + (0.889 − 3.89i)2-s + 5.19·3-s + (−14.4 − 6.93i)4-s + (4.61 − 20.2i)6-s + 44.0·7-s + (−39.8 + 50.0i)8-s + 27·9-s + 81.7i·11-s + (−74.9 − 36.0i)12-s + 192. i·13-s + (39.2 − 171. i)14-s + (159. + 199. i)16-s + 415. i·17-s + (24.0 − 105. i)18-s − 23.6i·19-s + ⋯
L(s)  = 1  + (0.222 − 0.974i)2-s + 0.577·3-s + (−0.901 − 0.433i)4-s + (0.128 − 0.562i)6-s + 0.899·7-s + (−0.622 + 0.782i)8-s + 0.333·9-s + 0.675i·11-s + (−0.520 − 0.250i)12-s + 1.13i·13-s + (0.200 − 0.877i)14-s + (0.624 + 0.781i)16-s + 1.43i·17-s + (0.0740 − 0.324i)18-s − 0.0655i·19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.999 - 0.0153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(300\)    =    \(2^{2} \cdot 3 \cdot 5^{2}\)
Sign: $0.999 - 0.0153i$
Analytic conductor: \(31.0109\)
Root analytic conductor: \(5.56875\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{300} (199, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 300,\ (\ :2),\ 0.999 - 0.0153i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.374535435\)
\(L(\frac12)\) \(\approx\) \(2.374535435\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

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

−11.06137650646370250657390108361, −10.31535181127301920492710464850, −9.251497244478983945637150364712, −8.526469363695402856631586981074, −7.45915125136558070158506123716, −5.92996319943846369404576107852, −4.55221554936107978054310885013, −3.89016856411536500162433918115, −2.26325205776344366958161433095, −1.48032237335619567755599106645, 0.66491607538276446393868775179, 2.72512389419101600145813731911, 4.02913450460899595064522755435, 5.15124854590643523850142717842, 6.08464433306232605226106766977, 7.56149948729560282570485448225, 7.938851078727209784936568557830, 8.951475556125906881168090765134, 9.861183286438137987786489502755, 11.13498071590888185567976905795

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