Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.385 + 0.922i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)4-s + (0.258 − 0.965i)7-s + (−1.22 − 1.22i)13-s + (0.499 − 0.866i)16-s + (−0.866 − 0.5i)19-s + (−0.258 − 0.965i)28-s + (1 + 1.73i)31-s + (0.448 + 1.67i)37-s + (−0.866 − 0.499i)49-s + (−1.67 − 0.448i)52-s + (0.5 − 0.866i)61-s − 0.999i·64-s + (1.67 + 0.448i)67-s + (−0.448 + 1.67i)73-s − 0.999·76-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)4-s + (0.258 − 0.965i)7-s + (−1.22 − 1.22i)13-s + (0.499 − 0.866i)16-s + (−0.866 − 0.5i)19-s + (−0.258 − 0.965i)28-s + (1 + 1.73i)31-s + (0.448 + 1.67i)37-s + (−0.866 − 0.499i)49-s + (−1.67 − 0.448i)52-s + (0.5 − 0.866i)61-s − 0.999i·64-s + (1.67 + 0.448i)67-s + (−0.448 + 1.67i)73-s − 0.999·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.385 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.385 + 0.922i$
motivic weight  =  \(0\)
character  :  $\chi_{1575} (793, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ 0.385 + 0.922i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.292459688\)
\(L(\frac12)\)  \(\approx\)  \(1.292459688\)
\(L(1)\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-0.258 + 0.965i)T \)
good2 \( 1 + (-0.866 + 0.5i)T^{2} \)
11 \( 1 + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (1.22 + 1.22i)T + iT^{2} \)
17 \( 1 + (0.866 + 0.5i)T^{2} \)
19 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 - 0.5i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.448 - 1.67i)T + (-0.866 + 0.5i)T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + (-0.866 + 0.5i)T^{2} \)
53 \( 1 + (-0.866 - 0.5i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 - 0.448i)T + (0.866 + 0.5i)T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (0.448 - 1.67i)T + (-0.866 - 0.5i)T^{2} \)
79 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 + iT^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (-1.22 + 1.22i)T - iT^{2} \)
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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

−9.951592925000635889525354481582, −8.494141219863861204969707475587, −7.78774483164192325258540193149, −6.98180430968360976672507201809, −6.44816496853024155598705855075, −5.25094658998857753458134835169, −4.66501967768058129842898324007, −3.27825115821248863519506582903, −2.39203838478409052843605799611, −1.03032890668133285385043912399, 2.08500688479143862743811439639, 2.43098139386542800302091404075, 3.83311240615520857392924160459, 4.74012550645780998212308089202, 5.89519199970182957966490877131, 6.50283566932256669455131674809, 7.44877139238149787483532861420, 8.059007735899930348440632505708, 8.981799427800578227010804549900, 9.657223612762648477945291476773

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