Properties

Degree 2
Conductor $ 2^{2} \cdot 3^{4} \cdot 5 $
Sign $-0.939 + 0.342i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)5-s + (−1 + 1.73i)7-s + (−1 − 1.73i)13-s − 6·17-s − 4·19-s + (−3 − 5.19i)23-s + (−0.499 + 0.866i)25-s + (−3 + 5.19i)29-s + (2 + 3.46i)31-s − 1.99·35-s + 2·37-s + (−3 − 5.19i)41-s + (5 − 8.66i)43-s + (3 − 5.19i)47-s + (1.50 + 2.59i)49-s + ⋯
L(s)  = 1  + (0.223 + 0.387i)5-s + (−0.377 + 0.654i)7-s + (−0.277 − 0.480i)13-s − 1.45·17-s − 0.917·19-s + (−0.625 − 1.08i)23-s + (−0.0999 + 0.173i)25-s + (−0.557 + 0.964i)29-s + (0.359 + 0.622i)31-s − 0.338·35-s + 0.328·37-s + (−0.468 − 0.811i)41-s + (0.762 − 1.32i)43-s + (0.437 − 0.757i)47-s + (0.214 + 0.371i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1620\)    =    \(2^{2} \cdot 3^{4} \cdot 5\)
\( \varepsilon \)  =  $-0.939 + 0.342i$
motivic weight  =  \(1\)
character  :  $\chi_{1620} (541, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 1620,\ (\ :1/2),\ -0.939 + 0.342i)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (1 - 1.73i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 + 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 6T + 17T^{2} \)
19 \( 1 + 4T + 19T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (3 + 5.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5 + 8.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 - 2T + 73T^{2} \)
79 \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3 - 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.879609668633676560438179903739, −8.518701332677239511880295642056, −7.30574166128665362043917886705, −6.57545210475335878147183798094, −5.90277677478490330798174406784, −4.92223288312488907651471559014, −3.93944715685640249462758706971, −2.76324459854099422021748940251, −2.02989984762113154748685340570, 0, 1.63169261711885065541736310622, 2.72807111839117201323872077617, 4.16105349651013886794664529191, 4.47514029994835495312362165577, 5.85851390667869065406076060278, 6.44791198724747539593326744519, 7.38160816732570281031336149602, 8.128006439029553491986246914026, 9.128268260075230639978810280679

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