Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{2} \cdot 7^{2} $
Sign $-0.876 + 0.480i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.629 − 1.26i)2-s + (−1.20 + 1.59i)4-s − 2.32i·5-s + (2.77 + 0.524i)8-s + (−2.94 + 1.46i)10-s − 3.58i·11-s + 2.93i·13-s + (−1.08 − 3.84i)16-s − 2.32i·17-s + 8.33·19-s + (3.71 + 2.80i)20-s + (−4.53 + 2.25i)22-s − 1.48i·23-s − 0.414·25-s + (3.71 − 1.84i)26-s + ⋯
L(s)  = 1  + (−0.445 − 0.895i)2-s + (−0.603 + 0.797i)4-s − 1.04i·5-s + (0.982 + 0.185i)8-s + (−0.931 + 0.463i)10-s − 1.07i·11-s + 0.812i·13-s + (−0.271 − 0.962i)16-s − 0.564i·17-s + 1.91·19-s + (0.829 + 0.628i)20-s + (−0.966 + 0.480i)22-s − 0.309i·23-s − 0.0828·25-s + (0.727 − 0.361i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.876 + 0.480i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1764\)    =    \(2^{2} \cdot 3^{2} \cdot 7^{2}\)
\( \varepsilon \)  =  $-0.876 + 0.480i$
motivic weight  =  \(1\)
character  :  $\chi_{1764} (1567, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1764,\ (\ :1/2),\ -0.876 + 0.480i)\)
\(L(1)\)  \(\approx\)  \(1.178142335\)
\(L(\frac12)\)  \(\approx\)  \(1.178142335\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.629 + 1.26i)T \)
3 \( 1 \)
7 \( 1 \)
good5 \( 1 + 2.32iT - 5T^{2} \)
11 \( 1 + 3.58iT - 11T^{2} \)
13 \( 1 - 2.93iT - 13T^{2} \)
17 \( 1 + 2.32iT - 17T^{2} \)
19 \( 1 - 8.33T + 19T^{2} \)
23 \( 1 + 1.48iT - 23T^{2} \)
29 \( 1 - 7.86T + 29T^{2} \)
31 \( 1 + 3.45T + 31T^{2} \)
37 \( 1 + 8.24T + 37T^{2} \)
41 \( 1 - 2.89iT - 41T^{2} \)
43 \( 1 + 6.37iT - 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 8.59T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 2.93iT - 61T^{2} \)
67 \( 1 + 9.02iT - 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 - 11.2iT - 73T^{2} \)
79 \( 1 + 15.3iT - 79T^{2} \)
83 \( 1 + 14.8T + 83T^{2} \)
89 \( 1 + 13.5iT - 89T^{2} \)
97 \( 1 - 5.35iT - 97T^{2} \)
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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

−8.891093246936704412727241888950, −8.592839370736878554402694791104, −7.64578400356047961743175022877, −6.73038537674487359279044752966, −5.34877771586099537551878771339, −4.84139191975332147598085315900, −3.72245473939323832177918032989, −2.90571027985495000956961917727, −1.53595776900806944372834322203, −0.59517265524206225672799480948, 1.30806444956721212805661182589, 2.75826620203352041647439127719, 3.82785206165265108736648119666, 5.04564594488092091591210661088, 5.64470734464818337976648445888, 6.79485173061306347931247497378, 7.10989710934573631268082319089, 7.912667278765075190979013198708, 8.684405246809536105005836572886, 9.872291913073605848165500975714

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