Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.876 - 0.481i$
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 − 0.585i·11-s + (0.0249 + 0.0249i)13-s − 1.00·14-s − 1.00·16-s + (−1.92 − 1.92i)17-s − 4.87i·19-s + (0.414 − 0.414i)22-s + (5.15 − 5.15i)23-s + 0.0352i·26-s + (−0.707 − 0.707i)28-s + 4.58·29-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 − 0.176i·11-s + (0.00691 + 0.00691i)13-s − 0.267·14-s − 0.250·16-s + (−0.466 − 0.466i)17-s − 1.11i·19-s + (0.0883 − 0.0883i)22-s + (1.07 − 1.07i)23-s + 0.00691i·26-s + (−0.133 − 0.133i)28-s + 0.850·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.876 - 0.481i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.876 - 0.481i)$
$L(1)$  $\approx$  $2.328250389$
$L(\frac12)$  $\approx$  $2.328250389$
$L(\frac{3}{2})$   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 \( 1 + 0.585iT - 11T^{2} \)
13 \( 1 + (-0.0249 - 0.0249i)T + 13iT^{2} \)
17 \( 1 + (1.92 + 1.92i)T + 17iT^{2} \)
19 \( 1 + 4.87iT - 19T^{2} \)
23 \( 1 + (-5.15 + 5.15i)T - 23iT^{2} \)
29 \( 1 - 4.58T + 29T^{2} \)
31 \( 1 - 9.83T + 31T^{2} \)
37 \( 1 + (2.87 - 2.87i)T - 37iT^{2} \)
41 \( 1 + 0.979iT - 41T^{2} \)
43 \( 1 + (-4.27 - 4.27i)T + 43iT^{2} \)
47 \( 1 + (-6.32 - 6.32i)T + 47iT^{2} \)
53 \( 1 + (-1.56 + 1.56i)T - 53iT^{2} \)
59 \( 1 - 0.670T + 59T^{2} \)
61 \( 1 - 2.35T + 61T^{2} \)
67 \( 1 + (5.08 - 5.08i)T - 67iT^{2} \)
71 \( 1 - 2.92iT - 71T^{2} \)
73 \( 1 + (2.51 + 2.51i)T + 73iT^{2} \)
79 \( 1 - 8.71iT - 79T^{2} \)
83 \( 1 + (-2.99 + 2.99i)T - 83iT^{2} \)
89 \( 1 - 9.72T + 89T^{2} \)
97 \( 1 + (4.74 - 4.74i)T - 97iT^{2} \)
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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

−8.706557437011765744595042210426, −7.990132591527217165835269008641, −6.93744980505266407006043120632, −6.62183814315565127039887362271, −5.73836137847209759865984225809, −4.78159091728159717368055073944, −4.37000381784387692670941718699, −2.99400679836436862778162835879, −2.57776330589408099818493215056, −0.793576430557116122452167975156, 0.945303765170667093007102829602, 2.05037955079324852595825770398, 3.08516076419862647878181931438, 3.85999792117470455866548747745, 4.63838241202694945842663202011, 5.51119551176214960986063133385, 6.27050557738348066589625407006, 7.01488452100201872263461745449, 7.85352304283140392450943848487, 8.736588916783181799080619199412

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