Properties

Label 2-8450-1.1-c1-0-61
Degree $2$
Conductor $8450$
Sign $1$
Analytic cond. $67.4735$
Root an. cond. $8.21423$
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 − 6-s + 3·7-s − 8-s − 2·9-s + 12-s − 3·14-s + 16-s + 3·17-s + 2·18-s − 6·19-s + 3·21-s − 6·23-s − 24-s − 5·27-s + 3·28-s − 32-s − 3·34-s − 2·36-s + 3·37-s + 6·38-s − 3·42-s − 43-s + 6·46-s + 3·47-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s + 1.13·7-s − 0.353·8-s − 2/3·9-s + 0.288·12-s − 0.801·14-s + 1/4·16-s + 0.727·17-s + 0.471·18-s − 1.37·19-s + 0.654·21-s − 1.25·23-s − 0.204·24-s − 0.962·27-s + 0.566·28-s − 0.176·32-s − 0.514·34-s − 1/3·36-s + 0.493·37-s + 0.973·38-s − 0.462·42-s − 0.152·43-s + 0.884·46-s + 0.437·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8450\)    =    \(2 \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(67.4735\)
Root analytic conductor: \(8.21423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{8450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.740245878\)
\(L(\frac12)\) \(\approx\) \(1.740245878\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 3 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 15 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−8.167268080397728825770144395167, −7.40862748149144466958550408179, −6.46656684018924522714660519369, −5.81919936521029094824073749797, −5.06391040756493978960151315681, −4.14226598568978309926244161894, −3.40448287769266831131199619471, −2.29391370673890045834033086930, −1.94991966120951197998042855087, −0.68349243649354398924238112786, 0.68349243649354398924238112786, 1.94991966120951197998042855087, 2.29391370673890045834033086930, 3.40448287769266831131199619471, 4.14226598568978309926244161894, 5.06391040756493978960151315681, 5.81919936521029094824073749797, 6.46656684018924522714660519369, 7.40862748149144466958550408179, 8.167268080397728825770144395167

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