Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.391 - 0.920i$
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 + 6.36i·11-s + (1.69 − 1.69i)13-s + 1.00·14-s − 1.00·16-s + (0.979 − 0.979i)17-s + (−4.49 − 4.49i)22-s + (−1.38 − 1.38i)23-s + 2.39i·26-s + (−0.707 + 0.707i)28-s + 5.81·29-s + 2.36·31-s + (0.707 − 0.707i)32-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 + 1.91i·11-s + (0.469 − 0.469i)13-s + 0.267·14-s − 0.250·16-s + (0.237 − 0.237i)17-s + (−0.959 − 0.959i)22-s + (−0.288 − 0.288i)23-s + 0.469i·26-s + (−0.133 + 0.133i)28-s + 1.08·29-s + 0.424·31-s + (0.125 − 0.125i)32-s + ⋯

Functional equation

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

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.391 - 0.920i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.391 - 0.920i)$
$L(1)$  $\approx$  $1.081579389$
$L(\frac12)$  $\approx$  $1.081579389$
$L(\frac{3}{2})$   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 \( 1 - 6.36iT - 11T^{2} \)
13 \( 1 + (-1.69 + 1.69i)T - 13iT^{2} \)
17 \( 1 + (-0.979 + 0.979i)T - 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + (1.38 + 1.38i)T + 23iT^{2} \)
29 \( 1 - 5.81T + 29T^{2} \)
31 \( 1 - 2.36T + 31T^{2} \)
37 \( 1 + (0.157 + 0.157i)T + 37iT^{2} \)
41 \( 1 - 8.94iT - 41T^{2} \)
43 \( 1 + (5.88 - 5.88i)T - 43iT^{2} \)
47 \( 1 + (-3.05 + 3.05i)T - 47iT^{2} \)
53 \( 1 + (-0.192 - 0.192i)T + 53iT^{2} \)
59 \( 1 + 10.3T + 59T^{2} \)
61 \( 1 + 0.979T + 61T^{2} \)
67 \( 1 + (9.88 + 9.88i)T + 67iT^{2} \)
71 \( 1 + 12.3iT - 71T^{2} \)
73 \( 1 + (4.71 - 4.71i)T - 73iT^{2} \)
79 \( 1 + 6.68iT - 79T^{2} \)
83 \( 1 + (-11.3 - 11.3i)T + 83iT^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + (-9.39 - 9.39i)T + 97iT^{2} \)
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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

−8.981325563676184879963801193816, −7.88501755299697275852117301745, −7.64980437278873359946000018155, −6.59069270204620985738710657225, −6.25978446431559035540151950825, −4.94369972229714537730241656318, −4.56920287934339171104987699934, −3.33352255449482713431928355951, −2.21284176914169607036193782658, −1.11088966471486031700894447246, 0.45524089846909760924220788309, 1.58778492385209691921281190557, 2.84431194767951064198483114211, 3.44706623867051132223828177442, 4.33941018801970184332099995642, 5.59087271006638524765072135265, 6.12776591061846178582847247607, 6.99117467939832645517695259924, 7.970445616031031707675299682205, 8.691837919638216396097280695264

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