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Properties

Label 2-248e2-1.1-c1-0-9
Degree $2$
Conductor $61504$
Sign $1$
Analytic cond. $491.111$
Root an. cond. $22.1610$
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  + 2·5-s − 3·9-s + 2·13-s + 6·17-s − 4·19-s − 8·23-s − 25-s + 2·29-s + 10·37-s − 6·41-s + 8·43-s − 6·45-s − 8·47-s − 7·49-s − 6·53-s + 12·59-s − 6·61-s + 4·65-s + 12·67-s + 8·71-s − 10·73-s + 8·79-s + 9·81-s + 8·83-s + 12·85-s + 6·89-s − 8·95-s + ⋯
L(s)  = 1  + 0.894·5-s − 9-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 1.66·23-s − 1/5·25-s + 0.371·29-s + 1.64·37-s − 0.937·41-s + 1.21·43-s − 0.894·45-s − 1.16·47-s − 49-s − 0.824·53-s + 1.56·59-s − 0.768·61-s + 0.496·65-s + 1.46·67-s + 0.949·71-s − 1.17·73-s + 0.900·79-s + 81-s + 0.878·83-s + 1.30·85-s + 0.635·89-s − 0.820·95-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 61504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(61504\)    =    \(2^{6} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(491.111\)
Root analytic conductor: \(22.1610\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 61504,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.274936340\)
\(L(\frac12)\) \(\approx\) \(2.274936340\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−14.34151309103945, −13.83213001913330, −13.35365220054425, −12.82414930107008, −12.24989029631141, −11.76686413985853, −11.25958485897620, −10.68741898422632, −10.15765595037748, −9.614753695745830, −9.372105537268707, −8.468097537812307, −8.057112360297133, −7.835446497356291, −6.737692965125393, −6.232185953253571, −5.919767400693519, −5.417338575922606, −4.760311383625453, −3.931123344525700, −3.443637143394505, −2.659840811455148, −2.114853207762413, −1.426544336743667, −0.5067663842404504, 0.5067663842404504, 1.426544336743667, 2.114853207762413, 2.659840811455148, 3.443637143394505, 3.931123344525700, 4.760311383625453, 5.417338575922606, 5.919767400693519, 6.232185953253571, 6.737692965125393, 7.835446497356291, 8.057112360297133, 8.468097537812307, 9.372105537268707, 9.614753695745830, 10.15765595037748, 10.68741898422632, 11.25958485897620, 11.76686413985853, 12.24989029631141, 12.82414930107008, 13.35365220054425, 13.83213001913330, 14.34151309103945

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