Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.952 - 0.304i$
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 + (0.717 + 2.54i)7-s − 0.999i·8-s + (−5.09 − 2.94i)11-s + 4.05i·13-s + (1.89 + 1.84i)14-s + (−0.5 − 0.866i)16-s + (−0.214 + 0.371i)17-s + (5.30 − 3.06i)19-s − 5.88·22-s + (−1.51 + 0.876i)23-s + (2.02 + 3.51i)26-s + (2.56 + 0.651i)28-s − 0.0419i·29-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + (0.271 + 0.962i)7-s − 0.353i·8-s + (−1.53 − 0.886i)11-s + 1.12i·13-s + (0.506 + 0.493i)14-s + (−0.125 − 0.216i)16-s + (−0.0520 + 0.0901i)17-s + (1.21 − 0.703i)19-s − 1.25·22-s + (−0.316 + 0.182i)23-s + (0.397 + 0.689i)26-s + (0.484 + 0.123i)28-s − 0.00778i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.952 - 0.304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.952 - 0.304i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1151, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.952 - 0.304i)$
$L(1)$  $\approx$  $2.462306895$
$L(\frac12)$  $\approx$  $2.462306895$
$L(\frac{3}{2})$   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 \( 1 + (-0.717 - 2.54i)T \)
good11 \( 1 + (5.09 + 2.94i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 - 4.05iT - 13T^{2} \)
17 \( 1 + (0.214 - 0.371i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.30 + 3.06i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.51 - 0.876i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.0419iT - 29T^{2} \)
31 \( 1 + (-7.92 - 4.57i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.536 - 0.928i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 8.61T + 41T^{2} \)
43 \( 1 - 11.0T + 43T^{2} \)
47 \( 1 + (0.481 + 0.834i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-11.3 - 6.57i)T + (26.5 + 45.8i)T^{2} \)
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 \( 1 + (6.32 - 10.9i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.54iT - 71T^{2} \)
73 \( 1 + (-8.08 - 4.66i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.35 + 9.28i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 + (-3.15 - 5.46i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2.59iT - 97T^{2} \)
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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

−8.846724800549487340565472403897, −7.946546165567423830834783457389, −7.23306488534981831312373289151, −6.15971316226089085112104490426, −5.59171951783330712523527694212, −4.93081250278763751311765452006, −4.08462460148599849952802809514, −2.76833608311407004361649049194, −2.56717752172423601713891424986, −1.09667375058078720060048103199, 0.70062067761310044573402922395, 2.25031851929147009398490803988, 3.10260627324005066507856257625, 4.07771650657139584263452512782, 4.85600680311984366528764807171, 5.47912606973822744305075579117, 6.28328946650811352043852261032, 7.43184457660118542669335002968, 7.68739153366789784631233562427, 8.186166151274157789641266681159

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