Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.833 + 0.552i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s + (2.54 + 0.732i)7-s + 0.999i·8-s + (2.07 + 1.19i)11-s − 5.67i·13-s + (−2.56 + 0.636i)14-s + (−0.5 − 0.866i)16-s + (−1.03 + 1.79i)17-s + (5.12 − 2.95i)19-s − 2.39·22-s + (1.61 − 0.930i)23-s + (2.83 + 4.91i)26-s + (1.90 − 1.83i)28-s − 4.88i·29-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.960 + 0.276i)7-s + 0.353i·8-s + (0.625 + 0.361i)11-s − 1.57i·13-s + (−0.686 + 0.170i)14-s + (−0.125 − 0.216i)16-s + (−0.251 + 0.435i)17-s + (1.17 − 0.678i)19-s − 0.511·22-s + (0.336 − 0.194i)23-s + (0.556 + 0.964i)26-s + (0.360 − 0.346i)28-s − 0.907i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.833 + 0.552i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.833 + 0.552i)$
$L(1)$  $\approx$  $1.539144032$
$L(\frac12)$  $\approx$  $1.539144032$
$L(\frac{3}{2})$   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 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.54 - 0.732i)T \)
good11 \( 1 + (-2.07 - 1.19i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + 5.67iT - 13T^{2} \)
17 \( 1 + (1.03 - 1.79i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.12 + 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.61 + 0.930i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.88iT - 29T^{2} \)
31 \( 1 + (3.92 + 2.26i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.48 + 2.57i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 7.04T + 41T^{2} \)
43 \( 1 + 8.55T + 43T^{2} \)
47 \( 1 + (2.78 + 4.83i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.62 + 2.09i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.00 - 1.73i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-10.7 + 6.22i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.81 + 6.60i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.14iT - 71T^{2} \)
73 \( 1 + (-0.937 - 0.541i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.38 + 14.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.6T + 83T^{2} \)
89 \( 1 + (-6.63 - 11.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 12.8iT - 97T^{2} \)
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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

−8.498543265546138749440803959335, −7.911076748786006278912781576311, −7.32250919739504855766682510848, −6.44826461880176266936532481749, −5.50087737690432244104933553350, −5.05505638483482525125852032794, −3.93607380292094878256201754707, −2.79906145581324389751655909587, −1.76125801086840007519565819667, −0.65970763423526714915751830692, 1.20382962922265318448250438928, 1.80278145915663562194900458943, 3.09733509987949056666947302464, 3.99384601063181791864523555068, 4.78237043771099362360336023876, 5.70866727317157361127399646640, 6.86747589031731882409037108296, 7.20469854083835107052208303359, 8.162600762322239836078222798506, 8.819683748321939161456095121632

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