Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.0618 + 0.998i$
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 + 4.82i·11-s + (3.41 − 3.41i)13-s − 1.00·14-s − 1.00·16-s + (1.58 − 1.58i)17-s − 7.07i·19-s + (3.41 + 3.41i)22-s + (4.82 + 4.82i)23-s − 4.82i·26-s + (−0.707 + 0.707i)28-s + 3.17·29-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 + 1.45i·11-s + (0.946 − 0.946i)13-s − 0.267·14-s − 0.250·16-s + (0.384 − 0.384i)17-s − 1.62i·19-s + (0.727 + 0.727i)22-s + (1.00 + 1.00i)23-s − 0.946i·26-s + (−0.133 + 0.133i)28-s + 0.588·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0618 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.0618 + 0.998i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.0618 + 0.998i)$
$L(1)$  $\approx$  $2.384424725$
$L(\frac12)$  $\approx$  $2.384424725$
$L(\frac{3}{2})$   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 \( 1 - 4.82iT - 11T^{2} \)
13 \( 1 + (-3.41 + 3.41i)T - 13iT^{2} \)
17 \( 1 + (-1.58 + 1.58i)T - 17iT^{2} \)
19 \( 1 + 7.07iT - 19T^{2} \)
23 \( 1 + (-4.82 - 4.82i)T + 23iT^{2} \)
29 \( 1 - 3.17T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 + (-6.07 - 6.07i)T + 37iT^{2} \)
41 \( 1 + 4.82iT - 41T^{2} \)
43 \( 1 + (6.41 - 6.41i)T - 43iT^{2} \)
47 \( 1 + (-6.41 + 6.41i)T - 47iT^{2} \)
53 \( 1 + (4.82 + 4.82i)T + 53iT^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 6.58T + 61T^{2} \)
67 \( 1 + (0.414 + 0.414i)T + 67iT^{2} \)
71 \( 1 + 12.2iT - 71T^{2} \)
73 \( 1 + (-6 + 6i)T - 73iT^{2} \)
79 \( 1 + 13.3iT - 79T^{2} \)
83 \( 1 + (9.65 + 9.65i)T + 83iT^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + (-8.24 - 8.24i)T + 97iT^{2} \)
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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

−8.663125778760956825318708407070, −7.52551959232140122987095346838, −7.05185970240835956127642219843, −6.18180175116231078942129121606, −5.15989531559612063626833718757, −4.74077126410742880052463871072, −3.61372810419164707125185692265, −2.98712003906362764588121473439, −1.87864392233619258633348698830, −0.71929279441015433816873017112, 1.13690673664451188197812815059, 2.51140432140499900854548179046, 3.61437243040363494040631559907, 3.96583913457493867434628979561, 5.24656732081034617183707814961, 5.93776017157023831897793970369, 6.36805917819107456720648880059, 7.25597637206329035206554533620, 8.259883815830986326462205742932, 8.619035416202324451951328664844

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