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 − 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

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.920 + 0.391i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.920 + 0.391i)$
$L(1)$  $\approx$  $0.4376629923$
$L(\frac12)$  $\approx$  $0.4376629923$
$L(\frac{3}{2})$   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

−8.262151072153941294537633600419, −7.86711772364214437270611431630, −6.70835436626810770826151219292, −6.23033658529723892148208032893, −5.55490016061135402921097099769, −4.54737160054187929776389584088, −3.52790090309716307278554287997, −2.69470080703496344550774034171, −1.19900010185303802796335760177, −0.16674120930113497476945888956, 1.77250929371305397557575413780, 2.02945982755519391596877403723, 3.69930049165607239535164115507, 3.92019096342043289128095081050, 5.19362360362796875150173218657, 6.02926774335826002564268288065, 6.96172268029782420757261247750, 7.54007130349361264460221074299, 8.371747837926617928925327338355, 9.189304375303610138554947134875

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