Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $-0.982 - 0.186i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (−2.64 − 0.0951i)7-s − 8-s − 5.28i·11-s − 2.19·13-s + (2.64 + 0.0951i)14-s + 16-s + 1.04i·17-s − 6.43i·19-s + 5.28i·22-s + 7.47·23-s + 2.19·26-s + (−2.64 − 0.0951i)28-s + 7.47i·29-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + (−0.999 − 0.0359i)7-s − 0.353·8-s − 1.59i·11-s − 0.607·13-s + (0.706 + 0.0254i)14-s + 0.250·16-s + 0.253i·17-s − 1.47i·19-s + 1.12i·22-s + 1.55·23-s + 0.429·26-s + (−0.499 − 0.0179i)28-s + 1.38i·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.982 - 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $-0.982 - 0.186i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (3149, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ -0.982 - 0.186i)$
$L(1)$  $\approx$  $0.2167560032$
$L(\frac12)$  $\approx$  $0.2167560032$
$L(\frac{3}{2})$   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 \( 1 + T \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 + (2.64 + 0.0951i)T \)
good11 \( 1 + 5.28iT - 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 - 1.04iT - 17T^{2} \)
19 \( 1 + 6.43iT - 19T^{2} \)
23 \( 1 - 7.47T + 23T^{2} \)
29 \( 1 - 7.47iT - 29T^{2} \)
31 \( 1 + 9.09iT - 31T^{2} \)
37 \( 1 - 0.855iT - 37T^{2} \)
41 \( 1 - 2.19T + 41T^{2} \)
43 \( 1 + 0.954iT - 43T^{2} \)
47 \( 1 - 11.0iT - 47T^{2} \)
53 \( 1 + 3.09T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 - 8.05iT - 61T^{2} \)
67 \( 1 + 5.33iT - 67T^{2} \)
71 \( 1 - 6.43iT - 71T^{2} \)
73 \( 1 + 4.57T + 73T^{2} \)
79 \( 1 + 15.6T + 79T^{2} \)
83 \( 1 + 4.38iT - 83T^{2} \)
89 \( 1 + 4.28T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−8.456246989889131524581580386160, −7.50500047182248762309500269887, −6.88505079597123975528613960797, −6.15701821214629007723426480861, −5.43532301720846821384179903633, −4.33016766888188145902847476431, −3.01686338833618637410252021198, −2.84612159361059767283120751754, −1.12224560123391708437908720527, −0.094314188738766299298412955579, 1.47288865494597226984905351152, 2.48922845705925419193822425508, 3.36932137167706376564718741870, 4.43798177089736015425056891710, 5.32792405072021427235231321450, 6.29517152787321049817042645667, 7.04025825690318944914603326019, 7.44659368747044618104563796266, 8.384279543524990020740803319890, 9.240556768624568506878709091006

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