Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.999 + 0.0387i$
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 + 3.41i·11-s + (1.43 − 1.43i)13-s − 1.00·14-s − 1.00·16-s + (−1.54 + 1.54i)17-s + 2.04i·19-s + (−2.41 − 2.41i)22-s + (−1.15 − 1.15i)23-s + 2.03i·26-s + (0.707 − 0.707i)28-s − 8.04·29-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 + 1.02i·11-s + (0.399 − 0.399i)13-s − 0.267·14-s − 0.250·16-s + (−0.374 + 0.374i)17-s + 0.470i·19-s + (−0.514 − 0.514i)22-s + (−0.241 − 0.241i)23-s + 0.399i·26-s + (0.133 − 0.133i)28-s − 1.49·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0387i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.999 + 0.0387i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (1457, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.999 + 0.0387i)$
$L(1)$  $\approx$  $0.5441789914$
$L(\frac12)$  $\approx$  $0.5441789914$
$L(\frac{3}{2})$   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 \( 1 - 3.41iT - 11T^{2} \)
13 \( 1 + (-1.43 + 1.43i)T - 13iT^{2} \)
17 \( 1 + (1.54 - 1.54i)T - 17iT^{2} \)
19 \( 1 - 2.04iT - 19T^{2} \)
23 \( 1 + (1.15 + 1.15i)T + 23iT^{2} \)
29 \( 1 + 8.04T + 29T^{2} \)
31 \( 1 + 6.37T + 31T^{2} \)
37 \( 1 + (0.0498 + 0.0498i)T + 37iT^{2} \)
41 \( 1 - 7.94iT - 41T^{2} \)
43 \( 1 + (-4.65 + 4.65i)T - 43iT^{2} \)
47 \( 1 + (1.39 - 1.39i)T - 47iT^{2} \)
53 \( 1 + (-2.97 - 2.97i)T + 53iT^{2} \)
59 \( 1 - 4.25T + 59T^{2} \)
61 \( 1 + 6.88T + 61T^{2} \)
67 \( 1 + (5.84 + 5.84i)T + 67iT^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (-5.97 + 5.97i)T - 73iT^{2} \)
79 \( 1 - 11.6iT - 79T^{2} \)
83 \( 1 + (1.52 + 1.52i)T + 83iT^{2} \)
89 \( 1 + 5.72T + 89T^{2} \)
97 \( 1 + (9.64 + 9.64i)T + 97iT^{2} \)
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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

−9.106559509147379875046107146915, −8.200149356891323633720819876863, −7.64556824346504073215962251898, −6.94214326743901751947937503748, −6.05199301048010566552965013535, −5.43495273634966490472205027368, −4.51845727995498976157705070520, −3.65016962567257992244009770729, −2.29683296931882682510921698734, −1.46866451412770780430076149114, 0.19918555011668512923708670613, 1.43772437434668039705318438539, 2.44862624316212779760033996747, 3.51835709902715486372016589097, 4.12725519028978230562365607079, 5.26214972060607768921405224695, 6.01937005867561688803697406990, 7.04277725481924682775353351875, 7.58733903249275815198059944674, 8.526827023669060855541856124007

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