Properties

Label 2-50430-1.1-c1-0-10
Degree $2$
Conductor $50430$
Sign $1$
Analytic cond. $402.685$
Root an. cond. $20.0670$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3-s + 4-s + 5-s + 6-s + 8-s + 9-s + 10-s − 4·11-s + 12-s + 2·13-s + 15-s + 16-s − 2·17-s + 18-s + 4·19-s + 20-s − 4·22-s + 8·23-s + 24-s + 25-s + 2·26-s + 27-s + 2·29-s + 30-s + 32-s − 4·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.447·5-s + 0.408·6-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 1.20·11-s + 0.288·12-s + 0.554·13-s + 0.258·15-s + 1/4·16-s − 0.485·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.852·22-s + 1.66·23-s + 0.204·24-s + 1/5·25-s + 0.392·26-s + 0.192·27-s + 0.371·29-s + 0.182·30-s + 0.176·32-s − 0.696·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(50430\)    =    \(2 \cdot 3 \cdot 5 \cdot 41^{2}\)
Sign: $1$
Analytic conductor: \(402.685\)
Root analytic conductor: \(20.0670\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 50430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(6.202823613\)
\(L(\frac12)\) \(\approx\) \(6.202823613\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 - T \)
41 \( 1 \)
good7 \( 1 + p T^{2} \) 1.7.a
11 \( 1 + 4 T + p T^{2} \) 1.11.e
13 \( 1 - 2 T + p T^{2} \) 1.13.ac
17 \( 1 + 2 T + p T^{2} \) 1.17.c
19 \( 1 - 4 T + p T^{2} \) 1.19.ae
23 \( 1 - 8 T + p T^{2} \) 1.23.ai
29 \( 1 - 2 T + p T^{2} \) 1.29.ac
31 \( 1 + p T^{2} \) 1.31.a
37 \( 1 - 6 T + p T^{2} \) 1.37.ag
43 \( 1 - 12 T + p T^{2} \) 1.43.am
47 \( 1 - 8 T + p T^{2} \) 1.47.ai
53 \( 1 + 6 T + p T^{2} \) 1.53.g
59 \( 1 + 4 T + p T^{2} \) 1.59.e
61 \( 1 + 2 T + p T^{2} \) 1.61.c
67 \( 1 + 12 T + p T^{2} \) 1.67.m
71 \( 1 + p T^{2} \) 1.71.a
73 \( 1 - 10 T + p T^{2} \) 1.73.ak
79 \( 1 + 8 T + p T^{2} \) 1.79.i
83 \( 1 - 4 T + p T^{2} \) 1.83.ae
89 \( 1 - 6 T + p T^{2} \) 1.89.ag
97 \( 1 - 14 T + p T^{2} \) 1.97.ao
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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

−14.39818431859722, −13.98054462173637, −13.38016607270447, −13.18003916607475, −12.66310711783701, −12.15787952986311, −11.28816801012904, −10.98001510045882, −10.48267529005919, −9.893754366737923, −9.133599239680479, −8.962356811009419, −8.065532590131538, −7.574000176418418, −7.205263557534746, −6.346721754267433, −5.974246673126924, −5.188450691725322, −4.837553908325307, −4.164332024078123, −3.353288230042658, −2.808062732414902, −2.449707150896666, −1.503791227147112, −0.7674036105234862, 0.7674036105234862, 1.503791227147112, 2.449707150896666, 2.808062732414902, 3.353288230042658, 4.164332024078123, 4.837553908325307, 5.188450691725322, 5.974246673126924, 6.346721754267433, 7.205263557534746, 7.574000176418418, 8.065532590131538, 8.962356811009419, 9.133599239680479, 9.893754366737923, 10.48267529005919, 10.98001510045882, 11.28816801012904, 12.15787952986311, 12.66310711783701, 13.18003916607475, 13.38016607270447, 13.98054462173637, 14.39818431859722

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