Properties

Degree 2
Conductor $ 2^{4} \cdot 3 \cdot 7^{2} $
Sign $0.605 - 0.795i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)9-s + (−3 − 5.19i)11-s + 2·13-s + (−2 + 3.46i)19-s + (−3 + 5.19i)23-s + (2.5 + 4.33i)25-s − 0.999·27-s + 6·29-s + (4 + 6.92i)31-s + (3 − 5.19i)33-s + (−1 + 1.73i)37-s + (1 + 1.73i)39-s + 12·41-s + 4·43-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.166 + 0.288i)9-s + (−0.904 − 1.56i)11-s + 0.554·13-s + (−0.458 + 0.794i)19-s + (−0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s − 0.192·27-s + 1.11·29-s + (0.718 + 1.24i)31-s + (0.522 − 0.904i)33-s + (−0.164 + 0.284i)37-s + (0.160 + 0.277i)39-s + 1.87·41-s + 0.609·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
\( \varepsilon \)  =  $0.605 - 0.795i$
motivic weight  =  \(1\)
character  :  $\chi_{2352} (961, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 2352,\ (\ :1/2),\ 0.605 - 0.795i)\)
\(L(1)\)  \(\approx\)  \(1.806578091\)
\(L(\frac12)\)  \(\approx\)  \(1.806578091\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3 + 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 - 5.19i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 6T + 29T^{2} \)
31 \( 1 + (-4 - 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 12T + 41T^{2} \)
43 \( 1 - 4T + 43T^{2} \)
47 \( 1 + (-6 + 10.3i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-4 - 6.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 6T + 71T^{2} \)
73 \( 1 + (-5 - 8.66i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 12T + 83T^{2} \)
89 \( 1 + (6 - 10.3i)T + (-44.5 - 77.0i)T^{2} \)
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

−8.948570264024871620121846425338, −8.373679238947412289545268438251, −7.83923337769409244194363015835, −6.73037548610876755722401835656, −5.75848343351756164234287376954, −5.32447292644377262542449593788, −4.10081317745311599115166427115, −3.37804046707988641401349354688, −2.55499019634426510586029306901, −1.04903925782933709581693766877, 0.71040944456387151011724551109, 2.27278853436506697571150202001, 2.64498024173241659152332272215, 4.24858645066785721866275906937, 4.65574677400007537791932302968, 5.93615149394874140145709010498, 6.58070181055153680044381197078, 7.43304609625792039376836339982, 8.017069253397462394286008874360, 8.778935401036831239947267251023

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