Properties

Degree 4
Conductor $ 2^{2} \cdot 7^{2} $
Sign $1$
Motivic weight 3
Primitive no
Self-dual yes
Analytic rank 0

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 3-s − 7·5-s + 2·6-s − 20·7-s − 8·8-s + 27·9-s − 14·10-s − 35·11-s + 132·13-s − 40·14-s − 7·15-s − 16·16-s − 59·17-s + 54·18-s − 137·19-s − 20·21-s − 70·22-s + 7·23-s − 8·24-s + 125·25-s + 264·26-s + 80·27-s + 212·29-s − 14·30-s − 75·31-s − 35·33-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.192·3-s − 0.626·5-s + 0.136·6-s − 1.07·7-s − 0.353·8-s + 9-s − 0.442·10-s − 0.959·11-s + 2.81·13-s − 0.763·14-s − 0.120·15-s − 1/4·16-s − 0.841·17-s + 0.707·18-s − 1.65·19-s − 0.207·21-s − 0.678·22-s + 0.0634·23-s − 0.0680·24-s + 25-s + 1.99·26-s + 0.570·27-s + 1.35·29-s − 0.0852·30-s − 0.434·31-s − 0.184·33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 196 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(4\)
\( N \)  =  \(196\)    =    \(2^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(3\)
character  :  induced by $\chi_{14} (1, \cdot )$
primitive  :  no
self-dual  :  yes
analytic rank  =  \(0\)
Selberg data  =  \((4,\ 196,\ (\ :3/2, 3/2),\ 1)\)
\(L(2)\)  \(\approx\)  \(1.15357\)
\(L(\frac12)\)  \(\approx\)  \(1.15357\)
\(L(\frac{5}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7\}$,\(F_p(T)\) is a polynomial of degree 4. If $p \in \{2,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 3.
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - p T + p^{2} T^{2} \)
7$C_2$ \( 1 + 20 T + p^{3} T^{2} \)
good3$C_2^2$ \( 1 - T - 26 T^{2} - p^{3} T^{3} + p^{6} T^{4} \)
5$C_2^2$ \( 1 + 7 T - 76 T^{2} + 7 p^{3} T^{3} + p^{6} T^{4} \)
11$C_2^2$ \( 1 + 35 T - 106 T^{2} + 35 p^{3} T^{3} + p^{6} T^{4} \)
13$C_2$ \( ( 1 - 66 T + p^{3} T^{2} )^{2} \)
17$C_2^2$ \( 1 + 59 T - 1432 T^{2} + 59 p^{3} T^{3} + p^{6} T^{4} \)
19$C_2^2$ \( 1 + 137 T + 11910 T^{2} + 137 p^{3} T^{3} + p^{6} T^{4} \)
23$C_2^2$ \( 1 - 7 T - 12118 T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \)
29$C_2$ \( ( 1 - 106 T + p^{3} T^{2} )^{2} \)
31$C_2^2$ \( 1 + 75 T - 24166 T^{2} + 75 p^{3} T^{3} + p^{6} T^{4} \)
37$C_2^2$ \( 1 + 11 T - 50532 T^{2} + 11 p^{3} T^{3} + p^{6} T^{4} \)
41$C_2$ \( ( 1 + 498 T + p^{3} T^{2} )^{2} \)
43$C_2$ \( ( 1 - 260 T + p^{3} T^{2} )^{2} \)
47$C_2^2$ \( 1 - 171 T - 74582 T^{2} - 171 p^{3} T^{3} + p^{6} T^{4} \)
53$C_2^2$ \( 1 - 417 T + 25012 T^{2} - 417 p^{3} T^{3} + p^{6} T^{4} \)
59$C_2^2$ \( 1 - 17 T - 205090 T^{2} - 17 p^{3} T^{3} + p^{6} T^{4} \)
61$C_2^2$ \( 1 + 51 T - 224380 T^{2} + 51 p^{3} T^{3} + p^{6} T^{4} \)
67$C_2^2$ \( 1 + 439 T - 108042 T^{2} + 439 p^{3} T^{3} + p^{6} T^{4} \)
71$C_2$ \( ( 1 + 784 T + p^{3} T^{2} )^{2} \)
73$C_2^2$ \( 1 + 295 T - 301992 T^{2} + 295 p^{3} T^{3} + p^{6} T^{4} \)
79$C_2^2$ \( 1 - 495 T - 248014 T^{2} - 495 p^{3} T^{3} + p^{6} T^{4} \)
83$C_2$ \( ( 1 - 932 T + p^{3} T^{2} )^{2} \)
89$C_2^2$ \( 1 - 873 T + 57160 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \)
97$C_2$ \( ( 1 + 290 T + p^{3} T^{2} )^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−19.36659132660275986647232450062, −18.84861829834191611688641591318, −18.42395395886323971363820162841, −17.73376855449346056640164989882, −16.38042466698445010529767813229, −16.03861877898491807075026006561, −15.47552069754499605822576797552, −15.00722107260579078507327498172, −13.54746730171127226533518152529, −13.38296499294512487966267915579, −12.82745157815110757261047381609, −11.97993937386865071289635495549, −10.60580380214991759514489550893, −10.54641437726097788540688220612, −8.839065203126092887000071158281, −8.452930461734506816688691865851, −6.86227351184222756791851788691, −6.17419372141557392180812905306, −4.45809912835375765151201974082, −3.44148627176067932895503895911, 3.44148627176067932895503895911, 4.45809912835375765151201974082, 6.17419372141557392180812905306, 6.86227351184222756791851788691, 8.452930461734506816688691865851, 8.839065203126092887000071158281, 10.54641437726097788540688220612, 10.60580380214991759514489550893, 11.97993937386865071289635495549, 12.82745157815110757261047381609, 13.38296499294512487966267915579, 13.54746730171127226533518152529, 15.00722107260579078507327498172, 15.47552069754499605822576797552, 16.03861877898491807075026006561, 16.38042466698445010529767813229, 17.73376855449346056640164989882, 18.42395395886323971363820162841, 18.84861829834191611688641591318, 19.36659132660275986647232450062

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