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Properties

Label 2-248430-1.1-c1-0-29
Degree $2$
Conductor $248430$
Sign $1$
Analytic cond. $1983.72$
Root an. cond. $44.5390$
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 − 12-s + 15-s + 16-s + 6·17-s − 18-s − 4·19-s − 20-s + 24-s + 25-s − 27-s − 6·29-s − 30-s − 4·31-s − 32-s − 6·34-s + 36-s − 2·37-s + 4·38-s + 40-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 − 0.288·12-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 0.917·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.182·30-s − 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 0.328·37-s + 0.648·38-s + 0.158·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 248430 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(248430\)    =    \(2 \cdot 3 \cdot 5 \cdot 7^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(1983.72\)
Root analytic conductor: \(44.5390\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 248430,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9191237385\)
\(L(\frac12)\) \(\approx\) \(0.9191237385\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 \)
good11 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−12.81016804570992, −12.35347493355206, −11.80529860125562, −11.44985618659117, −10.97410618866499, −10.51623097804709, −10.21336445926927, −9.575069446509129, −9.139726951675599, −8.767471853632753, −8.002230489079857, −7.691410218526238, −7.372548403607252, −6.764535470438973, −6.114870008164473, −5.870132507099912, −5.245582809382621, −4.677353421766184, −4.084975018782053, −3.469960587796239, −3.084189377459947, −2.161150958505329, −1.718608815125742, −0.9598933193959529, −0.3641840347841358, 0.3641840347841358, 0.9598933193959529, 1.718608815125742, 2.161150958505329, 3.084189377459947, 3.469960587796239, 4.084975018782053, 4.677353421766184, 5.245582809382621, 5.870132507099912, 6.114870008164473, 6.764535470438973, 7.372548403607252, 7.691410218526238, 8.002230489079857, 8.767471853632753, 9.139726951675599, 9.575069446509129, 10.21336445926927, 10.51623097804709, 10.97410618866499, 11.44985618659117, 11.80529860125562, 12.35347493355206, 12.81016804570992

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