Properties

Degree 2
Conductor $ 2 \cdot 5^{2} $
Sign $1$
Motivic weight 5
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 14·3-s + 16·4-s − 56·6-s + 158·7-s − 64·8-s − 47·9-s − 148·11-s + 224·12-s + 684·13-s − 632·14-s + 256·16-s + 2.04e3·17-s + 188·18-s + 2.22e3·19-s + 2.21e3·21-s + 592·22-s − 1.24e3·23-s − 896·24-s − 2.73e3·26-s − 4.06e3·27-s + 2.52e3·28-s − 270·29-s − 2.04e3·31-s − 1.02e3·32-s − 2.07e3·33-s − 8.19e3·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.898·3-s + 1/2·4-s − 0.635·6-s + 1.21·7-s − 0.353·8-s − 0.193·9-s − 0.368·11-s + 0.449·12-s + 1.12·13-s − 0.861·14-s + 1/4·16-s + 1.71·17-s + 0.136·18-s + 1.41·19-s + 1.09·21-s + 0.260·22-s − 0.491·23-s − 0.317·24-s − 0.793·26-s − 1.07·27-s + 0.609·28-s − 0.0596·29-s − 0.382·31-s − 0.176·32-s − 0.331·33-s − 1.21·34-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(50\)    =    \(2 \cdot 5^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(5\)
character  :  $\chi_{50} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 50,\ (\ :5/2),\ 1)\)
\(L(3)\)  \(\approx\)  \(1.80598\)
\(L(\frac12)\)  \(\approx\)  \(1.80598\)
\(L(\frac{7}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + p^{2} T \)
5 \( 1 \)
good3 \( 1 - 14 T + p^{5} T^{2} \)
7 \( 1 - 158 T + p^{5} T^{2} \)
11 \( 1 + 148 T + p^{5} T^{2} \)
13 \( 1 - 684 T + p^{5} T^{2} \)
17 \( 1 - 2048 T + p^{5} T^{2} \)
19 \( 1 - 2220 T + p^{5} T^{2} \)
23 \( 1 + 1246 T + p^{5} T^{2} \)
29 \( 1 + 270 T + p^{5} T^{2} \)
31 \( 1 + 2048 T + p^{5} T^{2} \)
37 \( 1 + 4372 T + p^{5} T^{2} \)
41 \( 1 + 2398 T + p^{5} T^{2} \)
43 \( 1 - 2294 T + p^{5} T^{2} \)
47 \( 1 + 10682 T + p^{5} T^{2} \)
53 \( 1 - 2964 T + p^{5} T^{2} \)
59 \( 1 + 39740 T + p^{5} T^{2} \)
61 \( 1 + 42298 T + p^{5} T^{2} \)
67 \( 1 - 32098 T + p^{5} T^{2} \)
71 \( 1 + 4248 T + p^{5} T^{2} \)
73 \( 1 - 30104 T + p^{5} T^{2} \)
79 \( 1 - 35280 T + p^{5} T^{2} \)
83 \( 1 + 27826 T + p^{5} T^{2} \)
89 \( 1 + 85210 T + p^{5} T^{2} \)
97 \( 1 + 97232 T + p^{5} T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−14.48894733048733978744819970322, −13.77684563564674653682899144955, −11.98569556409143960566713686926, −10.91476061918097024736032333103, −9.536697094708796313829082460504, −8.291327428170497872094901139815, −7.68859691366184980347896374583, −5.54626952739052738478306820479, −3.28499825800365520275696337731, −1.46276566744046286466573347456, 1.46276566744046286466573347456, 3.28499825800365520275696337731, 5.54626952739052738478306820479, 7.68859691366184980347896374583, 8.291327428170497872094901139815, 9.536697094708796313829082460504, 10.91476061918097024736032333103, 11.98569556409143960566713686926, 13.77684563564674653682899144955, 14.48894733048733978744819970322

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