Properties

Degree 2
Conductor $ 2^{4} \cdot 7 \cdot 17^{2} $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 7-s + 9-s − 4·13-s − 2·19-s − 2·21-s − 5·25-s + 4·27-s + 6·29-s − 4·31-s − 2·37-s + 8·39-s − 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s + 4·57-s + 6·59-s − 8·61-s + 63-s + 4·67-s − 2·73-s + 10·75-s + 8·79-s − 11·81-s + 6·83-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.10·13-s − 0.458·19-s − 0.436·21-s − 25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.328·37-s + 1.28·39-s − 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s + 0.529·57-s + 0.781·59-s − 1.02·61-s + 0.125·63-s + 0.488·67-s − 0.234·73-s + 1.15·75-s + 0.900·79-s − 1.22·81-s + 0.658·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(32368\)    =    \(2^{4} \cdot 7 \cdot 17^{2}\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{32368} (1, \cdot )$
Sato-Tate  :  $\mathrm{SU}(2)$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 32368,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;7,\;17\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;7,\;17\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−15.26272881518953, −14.89113096748579, −14.24967049524677, −13.72239902596623, −13.18434644610135, −12.30403017710645, −12.15788002311623, −11.70431473572361, −11.07013574289562, −10.56087889482214, −10.10510417175272, −9.570931276586979, −8.713971922191624, −8.345161402454639, −7.489142459129277, −7.078886987371320, −6.410367125615071, −5.858590903397252, −5.229737648703932, −4.845228980402497, −4.187234412870067, −3.380083826450902, −2.464917310380715, −1.838162648796834, −0.7894036805916308, 0, 0.7894036805916308, 1.838162648796834, 2.464917310380715, 3.380083826450902, 4.187234412870067, 4.845228980402497, 5.229737648703932, 5.858590903397252, 6.410367125615071, 7.078886987371320, 7.489142459129277, 8.345161402454639, 8.713971922191624, 9.570931276586979, 10.10510417175272, 10.56087889482214, 11.07013574289562, 11.70431473572361, 12.15788002311623, 12.30403017710645, 13.18434644610135, 13.72239902596623, 14.24967049524677, 14.89113096748579, 15.26272881518953

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