Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.998 - 0.0618i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.998 - 0.0618i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.998 - 0.0618i)$
$L(1)$  $\approx$  $1.527882631$
$L(\frac12)$  $\approx$  $1.527882631$
$L(\frac{3}{2})$   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

−8.997953168729009525212100117108, −8.221915661676878484168935332709, −7.28068147534567633367499586131, −6.87775136827822296247540039018, −6.09376516280642384009845611390, −5.13155162471317596276817384209, −4.51802361562843326412430100848, −3.73093390634685240299064505242, −2.60741457284700853663918369216, −1.69074683003654057948147483787, 0.38493884641783686940225514723, 1.50078223723858994580341304764, 2.88299047034554229672844615564, 3.44318323232005031188363056223, 4.20382489190269773402663113128, 5.37508497514264324980789053002, 5.87737869227204390068919823285, 6.59373826726488548006048449025, 7.58859316447461786757448219537, 8.490565478262868409094037698099

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