Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.988 - 0.153i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s + (−2.54 − 0.717i)7-s − 0.999·8-s + (−5.09 + 2.94i)11-s + 4.05·13-s + (−1.89 + 1.84i)14-s + (−0.5 + 0.866i)16-s + (−0.371 + 0.214i)17-s + (−5.30 − 3.06i)19-s + 5.88i·22-s + (0.876 − 1.51i)23-s + (2.02 − 3.51i)26-s + (0.651 + 2.56i)28-s − 0.0419i·29-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.962 − 0.271i)7-s − 0.353·8-s + (−1.53 + 0.886i)11-s + 1.12·13-s + (−0.506 + 0.493i)14-s + (−0.125 + 0.216i)16-s + (−0.0901 + 0.0520i)17-s + (−1.21 − 0.703i)19-s + 1.25i·22-s + (0.182 − 0.316i)23-s + (0.397 − 0.689i)26-s + (0.123 + 0.484i)28-s − 0.00778i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.153i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.988 - 0.153i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1349, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.988 - 0.153i)$
$L(1)$  $\approx$  $1.296741980$
$L(\frac12)$  $\approx$  $1.296741980$
$L(\frac{3}{2})$   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 \( 1 + (-0.5 + 0.866i)T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.54 + 0.717i)T \)
good11 \( 1 + (5.09 - 2.94i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 4.05T + 13T^{2} \)
17 \( 1 + (0.371 - 0.214i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.30 + 3.06i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.876 + 1.51i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0419iT - 29T^{2} \)
31 \( 1 + (-7.92 + 4.57i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.928 + 0.536i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 - 11.0iT - 43T^{2} \)
47 \( 1 + (-0.834 - 0.481i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.57 - 11.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.77 - 11.7i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.05 - 0.609i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.9 - 6.32i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + 2.54iT - 71T^{2} \)
73 \( 1 + (-4.66 - 8.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.35 + 9.28i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 10.1iT - 83T^{2} \)
89 \( 1 + (3.15 - 5.46i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 2.59T + 97T^{2} \)
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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

−8.816757437735764745179441127631, −8.004385336213393004200425883964, −7.14016618603056949263483501638, −6.28368346002039486472286499184, −5.71020719134966144993369251148, −4.52654110871430658038560005374, −4.13822229285877941384120349627, −2.83939938883491637181975728627, −2.44518297567055554712447998822, −0.901497623208167433432595597644, 0.44451454640367227495175950257, 2.28821911513956848597004415782, 3.23912962915868544016193281607, 3.85970470192378654486212007881, 5.01968802551127059607314140918, 5.76646916197183777385417733160, 6.27868009570827519500149593279, 7.00193551670167206136011866348, 8.088197937786961028505424315676, 8.436308406439179641438432193232

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