Properties

Degree 2
Conductor $ 2^{2} $
Sign $1$
Motivic weight 32
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 6.55e4·2-s + 4.29e9·4-s − 1.96e11·5-s + 2.81e14·8-s + 1.85e15·9-s − 1.28e16·10-s + 1.33e18·13-s + 1.84e19·16-s + 1.42e18·17-s + 1.21e20·18-s − 8.43e20·20-s + 1.53e22·25-s + 8.71e22·26-s + 4.62e23·29-s + 1.20e24·32-s + 9.35e22·34-s + 7.95e24·36-s + 1.33e25·37-s − 5.53e25·40-s − 1.17e26·41-s − 3.64e26·45-s + 1.10e27·49-s + 1.00e27·50-s + 5.71e27·52-s − 6.73e27·53-s + 3.03e28·58-s − 7.13e28·61-s + ⋯
L(s)  = 1  + 2-s + 4-s − 1.28·5-s + 8-s + 9-s − 1.28·10-s + 1.99·13-s + 16-s + 0.0293·17-s + 18-s − 1.28·20-s + 0.658·25-s + 1.99·26-s + 1.84·29-s + 32-s + 0.0293·34-s + 36-s + 1.08·37-s − 1.28·40-s − 1.84·41-s − 1.28·45-s + 49-s + 0.658·50-s + 1.99·52-s − 1.73·53-s + 1.84·58-s − 1.94·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(33-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4 ^{s/2} \, \Gamma_{\C}(s+16) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4\)    =    \(2^{2}\)
\( \varepsilon \)  =  $1$
motivic weight  =  \(32\)
character  :  $\chi_{4} (3, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  0
Selberg data  =  $(2,\ 4,\ (\ :16),\ 1)$
$L(\frac{33}{2})$  $\approx$  $3.642426946$
$L(\frac12)$  $\approx$  $3.642426946$
$L(17)$   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 \( 1 - p^{16} T \)
good3 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
5 \( 1 + 196496109694 T + p^{32} T^{2} \)
7 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
11 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
13 \( 1 - 1330087744899070082 T + p^{32} T^{2} \)
17 \( 1 - 1427124567881986562 T + p^{32} T^{2} \)
19 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
23 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
29 \( 1 - \)\(46\!\cdots\!42\)\( T + p^{32} T^{2} \)
31 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
37 \( 1 - \)\(13\!\cdots\!82\)\( T + p^{32} T^{2} \)
41 \( 1 + \)\(11\!\cdots\!18\)\( T + p^{32} T^{2} \)
43 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
47 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
53 \( 1 + \)\(67\!\cdots\!58\)\( T + p^{32} T^{2} \)
59 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
61 \( 1 + \)\(71\!\cdots\!78\)\( T + p^{32} T^{2} \)
67 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
71 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
73 \( 1 - \)\(60\!\cdots\!22\)\( T + p^{32} T^{2} \)
79 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
83 \( ( 1 - p^{16} T )( 1 + p^{16} T ) \)
89 \( 1 - \)\(17\!\cdots\!22\)\( T + p^{32} T^{2} \)
97 \( 1 - \)\(84\!\cdots\!42\)\( T + p^{32} T^{2} \)
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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

−16.09879542914653689273925326280, −15.40421836666852669723784990360, −13.52227661061581067796142581128, −12.08935896032832958751486962476, −10.78325193234564459703207053847, −8.038811212657539665648054455994, −6.50955274102792603425490742113, −4.45261341870275797506795180346, −3.41943780959080279307502149722, −1.21085702967646848479982126413, 1.21085702967646848479982126413, 3.41943780959080279307502149722, 4.45261341870275797506795180346, 6.50955274102792603425490742113, 8.038811212657539665648054455994, 10.78325193234564459703207053847, 12.08935896032832958751486962476, 13.52227661061581067796142581128, 15.40421836666852669723784990360, 16.09879542914653689273925326280

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