Properties

Degree 2
Conductor $ 2^{6} \cdot 3^{2} \cdot 7 $
Sign $0.169 - 0.985i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·7-s − 4.24i·11-s + 6i·13-s + 4.24·23-s − 5·25-s + 4.24·29-s + 4i·31-s − 6i·37-s + 8.48i·41-s − 6·43-s + 8.48·47-s − 49-s − 12.7·53-s + 8.48i·59-s − 6i·61-s + ⋯
L(s)  = 1  + 0.377i·7-s − 1.27i·11-s + 1.66i·13-s + 0.884·23-s − 25-s + 0.787·29-s + 0.718i·31-s − 0.986i·37-s + 1.32i·41-s − 0.914·43-s + 1.23·47-s − 0.142·49-s − 1.74·53-s + 1.10i·59-s − 0.768i·61-s + ⋯

Functional equation

\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned} \]
\[\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned} \]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(4032\)    =    \(2^{6} \cdot 3^{2} \cdot 7\)
\( \varepsilon \)  =  $0.169 - 0.985i$
motivic weight  =  \(1\)
character  :  $\chi_{4032} (2591, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 4032,\ (\ :1/2),\ 0.169 - 0.985i)$
$L(1)$  $\approx$  $1.495257923$
$L(\frac12)$  $\approx$  $1.495257923$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \] where, for $p \notin \{2,\;3,\;7\}$, \(F_p\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 - iT \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 4.24iT - 11T^{2} \)
13 \( 1 - 6iT - 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 4.24T + 23T^{2} \)
29 \( 1 - 4.24T + 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 6iT - 37T^{2} \)
41 \( 1 - 8.48iT - 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 - 8.48T + 47T^{2} \)
53 \( 1 + 12.7T + 53T^{2} \)
59 \( 1 - 8.48iT - 59T^{2} \)
61 \( 1 + 6iT - 61T^{2} \)
67 \( 1 - 12T + 67T^{2} \)
71 \( 1 - 4.24T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 - 10iT - 79T^{2} \)
83 \( 1 - 16.9iT - 83T^{2} \)
89 \( 1 - 8.48iT - 89T^{2} \)
97 \( 1 + 10T + 97T^{2} \)
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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

−8.582126208714631061290678870827, −8.055724235078652438779737146408, −6.97758220456534074887444198307, −6.45728967754517725464777800557, −5.69141735556805144807283262511, −4.86784729279691208672959611053, −4.00705067315927215019143456888, −3.16931585724086181994128067915, −2.21319651930375439917683443574, −1.11474261491534579866124691249, 0.46918899259827702462309735149, 1.72993651638594484156621580269, 2.77943286531179220805733196495, 3.61746058591613972159331306048, 4.58598802393403868999780784986, 5.20548079281903520116489852706, 6.06525497686410357034758676150, 6.91973077240824308686450639651, 7.60895450742240635917104821739, 8.102515148237133141706101506263

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