Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.309 - 0.950i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + (2.12 − 1.58i)7-s + i·8-s + 1.41i·11-s + 3.16i·13-s + (−1.58 − 2.12i)14-s + 16-s − 4.47·17-s + 1.41·22-s + 6i·23-s + 3.16·26-s + (−2.12 + 1.58i)28-s + 2.82i·29-s i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + (0.801 − 0.597i)7-s + 0.353i·8-s + 0.426i·11-s + 0.877i·13-s + (−0.422 − 0.566i)14-s + 0.250·16-s − 1.08·17-s + 0.301·22-s + 1.25i·23-s + 0.620·26-s + (−0.400 + 0.298i)28-s + 0.525i·29-s − 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.309 - 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.309 - 0.950i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (251, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.309 - 0.950i)$
$L(1)$  $\approx$  $0.9093473441$
$L(\frac12)$  $\approx$  $0.9093473441$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (-2.12 + 1.58i)T \)
good11 \( 1 - 1.41iT - 11T^{2} \)
13 \( 1 - 3.16iT - 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 4.24T + 37T^{2} \)
41 \( 1 + 9.48T + 41T^{2} \)
43 \( 1 + 8.48T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 - 9.48T + 59T^{2} \)
61 \( 1 - 13.4iT - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 - 5.65iT - 71T^{2} \)
73 \( 1 + 6.32iT - 73T^{2} \)
79 \( 1 - 4T + 79T^{2} \)
83 \( 1 + 8.94T + 83T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + 12.6iT - 97T^{2} \)
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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

−8.790326492929052579661291597052, −8.333343723581300875638228525634, −7.21926009475423263146366344584, −6.84402298729645556629080897001, −5.56077836872511160154322366381, −4.76304671383827737858832066118, −4.16941980832217075306604336860, −3.28973003205975117040239598282, −2.02416664421675885721315756439, −1.41810944225660202738530486471, 0.27390117608502847315366615123, 1.81882865060693184619663308319, 2.88894076939788356151535083180, 3.97328138668293698745903473401, 4.97222725885207290654678758740, 5.35423191107027993362325851204, 6.40827197693951956320761867116, 6.86188085313094879766607364070, 8.163848677187609331124557371588, 8.227275144805085392486336367891

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