Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \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 + (−0.707 + 0.707i)7-s + (−0.707 + 0.707i)8-s + 0.828i·11-s + (−0.585 − 0.585i)13-s − 1.00·14-s − 1.00·16-s + (−4.41 − 4.41i)17-s − 7.07i·19-s + (−0.585 + 0.585i)22-s + (0.828 − 0.828i)23-s − 0.828i·26-s + (−0.707 − 0.707i)28-s + 8.82·29-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 + 0.249i·11-s + (−0.162 − 0.162i)13-s − 0.267·14-s − 0.250·16-s + (−1.07 − 1.07i)17-s − 1.62i·19-s + (−0.124 + 0.124i)22-s + (0.172 − 0.172i)23-s − 0.162i·26-s + (−0.133 − 0.133i)28-s + 1.63·29-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.920 + 0.391i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.920 + 0.391i)$
$L(1)$  $\approx$  $1.849962593$
$L(\frac12)$  $\approx$  $1.849962593$
$L(\frac{3}{2})$   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 \( 1 - 0.828iT - 11T^{2} \)
13 \( 1 + (0.585 + 0.585i)T + 13iT^{2} \)
17 \( 1 + (4.41 + 4.41i)T + 17iT^{2} \)
19 \( 1 + 7.07iT - 19T^{2} \)
23 \( 1 + (-0.828 + 0.828i)T - 23iT^{2} \)
29 \( 1 - 8.82T + 29T^{2} \)
31 \( 1 + 3.17T + 31T^{2} \)
37 \( 1 + (-8.07 + 8.07i)T - 37iT^{2} \)
41 \( 1 + 0.828iT - 41T^{2} \)
43 \( 1 + (-3.58 - 3.58i)T + 43iT^{2} \)
47 \( 1 + (3.58 + 3.58i)T + 47iT^{2} \)
53 \( 1 + (0.828 - 0.828i)T - 53iT^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 9.41T + 61T^{2} \)
67 \( 1 + (2.41 - 2.41i)T - 67iT^{2} \)
71 \( 1 - 3.75iT - 71T^{2} \)
73 \( 1 + (6 + 6i)T + 73iT^{2} \)
79 \( 1 + 9.31iT - 79T^{2} \)
83 \( 1 + (1.65 - 1.65i)T - 83iT^{2} \)
89 \( 1 + 9.31T + 89T^{2} \)
97 \( 1 + (-0.242 + 0.242i)T - 97iT^{2} \)
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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

−8.720556899339470689910797065539, −7.70437195534241018625052617316, −6.96143340519452991967018114441, −6.53085411447327993439922401296, −5.51710208787576701465028035149, −4.78738224738687406590127760567, −4.18556990125430740096454471695, −2.89271933453093806243905262050, −2.39328013354049563204920864410, −0.51920011179929926720435907178, 1.14598137109386565445080191204, 2.18883017624833396015244947753, 3.20367700118773287726673881381, 4.03124987318200137611955783971, 4.65176934704160784243273241255, 5.75800527226934302478446549260, 6.30597431908173016635398092868, 7.06829328124899939097360243426, 8.169355393188612734992913253779, 8.626401710329880428236030551693

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