Properties

Degree 2
Conductor $ 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.894 - 0.447i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4-s + i·7-s + 16-s + i·28-s + 2i·37-s − 2i·43-s − 49-s + 64-s − 2i·67-s − 2·79-s + 2·109-s + i·112-s + ⋯
L(s)  = 1  + 4-s + i·7-s + 16-s + i·28-s + 2i·37-s − 2i·43-s − 49-s + 64-s − 2i·67-s − 2·79-s + 2·109-s + i·112-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(1575\)    =    \(3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 - 0.447i$
motivic weight  =  \(0\)
character  :  $\chi_{1575} (874, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 1575,\ (\ :0),\ 0.894 - 0.447i)\)
\(L(\frac{1}{2})\)  \(\approx\)  \(1.405897070\)
\(L(\frac12)\)  \(\approx\)  \(1.405897070\)
\(L(1)\)   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 \( 1 - iT \)
good2 \( 1 - T^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 2iT - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2iT - T^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 2T + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−9.751939668581427896467730196113, −8.785094445161926091883884528579, −8.120982520058901001726358376490, −7.20254116341639988099969785528, −6.42567958369940700772505248296, −5.72494990932537743023290213341, −4.85669050903742721122450969610, −3.47715774174445045269970651000, −2.61732237927340724392963698213, −1.66337124634595529707566199574, 1.27033692199437885483573765158, 2.49888726773389951467309249421, 3.53510049815431303241984854106, 4.44484030762349950054090385695, 5.62711434928389653058091435628, 6.41633951449973542666162314434, 7.25543411897011803892344680586, 7.69533806277893102285417128286, 8.696821163695744809947302107927, 9.777662703165187287566671656296

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