Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.0301 + 0.999i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0301 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0301 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.0301 + 0.999i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1601, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.0301 + 0.999i)$
$L(1)$  $\approx$  $1.663331486$
$L(\frac12)$  $\approx$  $1.663331486$
$L(\frac{3}{2})$   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

−8.437004505883030708037260124086, −7.87584815289856735564049611173, −7.19644517422028535775900262206, −6.32095864623962929676621988917, −5.58159877579507622790077532258, −4.37145051907641398905729116267, −3.61856815714198408372056191904, −2.92351477434401887998453155977, −1.36324395623809716247106396065, −0.790087059248403386856093662596, 1.21297624601716322453184964093, 2.04399574345065921016766935000, 3.15819680943805397563810044036, 4.43271749242663515119441830151, 5.00739325642959844731705379236, 6.06147782626485519407473952778, 6.78050223887803528244648222110, 7.19901772821135440571251246350, 8.322425464026757397227346247322, 8.874466418865673510002974097322

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