Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.391 + 0.920i$
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 + (−0.707 + 0.707i)7-s + (0.707 − 0.707i)8-s + 1.77i·11-s + (−0.692 − 0.692i)13-s + 1.00·14-s − 1.00·16-s + (−2.39 − 2.39i)17-s + (1.25 − 1.25i)22-s + (3.38 − 3.38i)23-s + 0.979i·26-s + (−0.707 − 0.707i)28-s + 4.42·29-s − 5.77·31-s + (0.707 + 0.707i)32-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.267 + 0.267i)7-s + (0.250 − 0.250i)8-s + 0.536i·11-s + (−0.192 − 0.192i)13-s + 0.267·14-s − 0.250·16-s + (−0.580 − 0.580i)17-s + (0.268 − 0.268i)22-s + (0.705 − 0.705i)23-s + 0.192i·26-s + (−0.133 − 0.133i)28-s + 0.821·29-s − 1.03·31-s + (0.125 + 0.125i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.391 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.391 + 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.391 + 0.920i)$
$L(1)$  $\approx$  $0.8283488324$
$L(\frac12)$  $\approx$  $0.8283488324$
$L(\frac{3}{2})$   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 \( 1 \)
7 \( 1 + (0.707 - 0.707i)T \)
good11 \( 1 - 1.77iT - 11T^{2} \)
13 \( 1 + (0.692 + 0.692i)T + 13iT^{2} \)
17 \( 1 + (2.39 + 2.39i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (-3.38 + 3.38i)T - 23iT^{2} \)
29 \( 1 - 4.42T + 29T^{2} \)
31 \( 1 + 5.77T + 31T^{2} \)
37 \( 1 + (5.91 - 5.91i)T - 37iT^{2} \)
41 \( 1 + 0.807iT - 41T^{2} \)
43 \( 1 + (-4.64 - 4.64i)T + 43iT^{2} \)
47 \( 1 + (7.47 + 7.47i)T + 47iT^{2} \)
53 \( 1 + (-3.56 + 3.56i)T - 53iT^{2} \)
59 \( 1 - 5.89T + 59T^{2} \)
61 \( 1 - 2.39T + 61T^{2} \)
67 \( 1 + (-0.641 + 0.641i)T - 67iT^{2} \)
71 \( 1 + 10.6iT - 71T^{2} \)
73 \( 1 + (5.70 + 5.70i)T + 73iT^{2} \)
79 \( 1 + 16.3iT - 79T^{2} \)
83 \( 1 + (0.171 - 0.171i)T - 83iT^{2} \)
89 \( 1 - 13.2T + 89T^{2} \)
97 \( 1 + (12.6 - 12.6i)T - 97iT^{2} \)
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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

−8.680857255839507685335354276107, −7.78230142340953683325279269597, −6.99376490626342140836333331994, −6.42031925943431416667988638743, −5.20425525635692083705114372139, −4.58044125887515865785113855032, −3.46681618673785245254679224481, −2.66484219710249802420293049454, −1.75261877867588580424357219594, −0.34724157076307811648912964621, 1.03762019743924534077857832254, 2.24079904223674948345022594074, 3.41228755982244574129217258589, 4.28204143024171523979981193867, 5.31095783076481907880738184930, 5.94931584399218244993680362340, 6.86940488870952465529325487940, 7.29194332971047641637046074440, 8.270130907458926871251101285243, 8.834739666744061995435576412230

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