Properties

Degree 2
Conductor $ 2^{2} $
Sign $1$
Motivic weight 4
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·2-s + 16·4-s − 14·5-s − 64·8-s + 81·9-s + 56·10-s − 238·13-s + 256·16-s + 322·17-s − 324·18-s − 224·20-s − 429·25-s + 952·26-s + 82·29-s − 1.02e3·32-s − 1.28e3·34-s + 1.29e3·36-s + 2.16e3·37-s + 896·40-s − 3.03e3·41-s − 1.13e3·45-s + 2.40e3·49-s + 1.71e3·50-s − 3.80e3·52-s + 2.48e3·53-s − 328·58-s − 6.95e3·61-s + ⋯
L(s)  = 1  − 2-s + 4-s − 0.559·5-s − 8-s + 9-s + 0.559·10-s − 1.40·13-s + 16-s + 1.11·17-s − 18-s − 0.559·20-s − 0.686·25-s + 1.40·26-s + 0.0975·29-s − 32-s − 1.11·34-s + 36-s + 1.57·37-s + 0.559·40-s − 1.80·41-s − 0.559·45-s + 49-s + 0.686·50-s − 1.40·52-s + 0.883·53-s − 0.0975·58-s − 1.86·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(4\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4,\ (\ :2),\ 1)$
$L(\frac{5}{2})$  $\approx$  $0.520074$
$L(\frac12)$  $\approx$  $0.520074$
$L(3)$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \neq 2$, \(F_p\) is a polynomial of degree 2. If $p = 2$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 + p^{2} T \)
good3 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
5 \( 1 + 14 T + p^{4} T^{2} \)
7 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
11 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
13 \( 1 + 238 T + p^{4} T^{2} \)
17 \( 1 - 322 T + p^{4} T^{2} \)
19 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
23 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
29 \( 1 - 82 T + p^{4} T^{2} \)
31 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
37 \( 1 - 2162 T + p^{4} T^{2} \)
41 \( 1 + 3038 T + p^{4} T^{2} \)
43 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
47 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
53 \( 1 - 2482 T + p^{4} T^{2} \)
59 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
61 \( 1 + 6958 T + p^{4} T^{2} \)
67 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
71 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
73 \( 1 - 1442 T + p^{4} T^{2} \)
79 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
83 \( ( 1 - p^{2} T )( 1 + p^{2} T ) \)
89 \( 1 + 9758 T + p^{4} T^{2} \)
97 \( 1 + 1918 T + p^{4} T^{2} \)
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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

−25.10576492074504075178503912426, −23.79530901106330221372802435825, −21.52319990358057348244285814727, −19.80599419980979259865604728242, −18.55479937229342170131233720500, −16.76273084018153415905874995989, −15.19455742605285905728438816594, −12.10283590839698097617567125444, −9.948419581278415768829474923694, −7.53515607601781601214992579901, 7.53515607601781601214992579901, 9.948419581278415768829474923694, 12.10283590839698097617567125444, 15.19455742605285905728438816594, 16.76273084018153415905874995989, 18.55479937229342170131233720500, 19.80599419980979259865604728242, 21.52319990358057348244285814727, 23.79530901106330221372802435825, 25.10576492074504075178503912426

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