Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $0.920 - 0.391i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 + 0.707i)2-s − 1.00i·4-s + (−2 + i)5-s + (−0.707 − 0.707i)7-s + (0.707 + 0.707i)8-s + (0.707 − 2.12i)10-s − 0.585i·11-s + (4 − 4i)13-s + 1.00·14-s − 1.00·16-s + (0.585 − 0.585i)17-s + 2.82i·19-s + (1.00 + 2.00i)20-s + (0.414 + 0.414i)22-s + (4.82 + 4.82i)23-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s − 0.500i·4-s + (−0.894 + 0.447i)5-s + (−0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + (0.223 − 0.670i)10-s − 0.176i·11-s + (1.10 − 1.10i)13-s + 0.267·14-s − 0.250·16-s + (0.142 − 0.142i)17-s + 0.648i·19-s + (0.223 + 0.447i)20-s + (0.0883 + 0.0883i)22-s + (1.00 + 1.00i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.920 - 0.391i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (197, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ 0.920 - 0.391i)\)
\(L(1)\)  \(\approx\)  \(0.948715 + 0.193174i\)
\(L(\frac12)\)  \(\approx\)  \(0.948715 + 0.193174i\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 \)
5 \( 1 + (2 - i)T \)
7 \( 1 + (0.707 + 0.707i)T \)
good11 \( 1 + 0.585iT - 11T^{2} \)
13 \( 1 + (-4 + 4i)T - 13iT^{2} \)
17 \( 1 + (-0.585 + 0.585i)T - 17iT^{2} \)
19 \( 1 - 2.82iT - 19T^{2} \)
23 \( 1 + (-4.82 - 4.82i)T + 23iT^{2} \)
29 \( 1 - 0.828T + 29T^{2} \)
31 \( 1 - 1.75T + 31T^{2} \)
37 \( 1 + (-6.24 - 6.24i)T + 37iT^{2} \)
41 \( 1 - 3.17iT - 41T^{2} \)
43 \( 1 + (-6.07 + 6.07i)T - 43iT^{2} \)
47 \( 1 + (-9.24 + 9.24i)T - 47iT^{2} \)
53 \( 1 + (2.58 + 2.58i)T + 53iT^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 9.89T + 61T^{2} \)
67 \( 1 + (-8.41 - 8.41i)T + 67iT^{2} \)
71 \( 1 + 1.17iT - 71T^{2} \)
73 \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \)
79 \( 1 - 5.65iT - 79T^{2} \)
83 \( 1 + (0.828 + 0.828i)T + 83iT^{2} \)
89 \( 1 - 3.17T + 89T^{2} \)
97 \( 1 + (4.58 + 4.58i)T + 97iT^{2} \)
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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

−10.65301883132749207915944434617, −9.810806631948365861928345303126, −8.683990708971765917490205500909, −7.987927346743529696102270542461, −7.25988681917535673205037516240, −6.31032236255564282041450638292, −5.36830266801142476087593288412, −3.95854093554322050856320098167, −3.03129177528224897245582911580, −0.906476570888264820172288205241, 0.993754232869376936238642401316, 2.63786456719835176599499616725, 3.89651375802012117561995202396, 4.66932376655048733606023765884, 6.18437178795575471796420523610, 7.15779085028621566952605844419, 8.109605047898240396814859249827, 8.989207279317772114346231041723, 9.370534574105661685746341501087, 10.91702289860776923312936267692

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