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 + 0.585i·11-s + (−0.0249 − 0.0249i)13-s + 1.00·14-s − 1.00·16-s + (−1.92 − 1.92i)17-s − 4.87i·19-s + (−0.414 + 0.414i)22-s + (5.15 − 5.15i)23-s − 0.0352i·26-s + (0.707 + 0.707i)28-s − 4.58·29-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 + 0.176i·11-s + (−0.00691 − 0.00691i)13-s + 0.267·14-s − 0.250·16-s + (−0.466 − 0.466i)17-s − 1.11i·19-s + (−0.0883 + 0.0883i)22-s + (1.07 − 1.07i)23-s − 0.00691i·26-s + (0.133 + 0.133i)28-s − 0.850·29-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  :  $\chi_{3150} (2843, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.999 + 0.0387i)$
$L(1)$  $\approx$  $2.378960570$
$L(\frac12)$  $\approx$  $2.378960570$
$L(\frac{3}{2})$   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 \( 1 - 0.585iT - 11T^{2} \)
13 \( 1 + (0.0249 + 0.0249i)T + 13iT^{2} \)
17 \( 1 + (1.92 + 1.92i)T + 17iT^{2} \)
19 \( 1 + 4.87iT - 19T^{2} \)
23 \( 1 + (-5.15 + 5.15i)T - 23iT^{2} \)
29 \( 1 + 4.58T + 29T^{2} \)
31 \( 1 - 9.83T + 31T^{2} \)
37 \( 1 + (-2.87 + 2.87i)T - 37iT^{2} \)
41 \( 1 - 0.979iT - 41T^{2} \)
43 \( 1 + (4.27 + 4.27i)T + 43iT^{2} \)
47 \( 1 + (-6.32 - 6.32i)T + 47iT^{2} \)
53 \( 1 + (-1.56 + 1.56i)T - 53iT^{2} \)
59 \( 1 + 0.670T + 59T^{2} \)
61 \( 1 - 2.35T + 61T^{2} \)
67 \( 1 + (-5.08 + 5.08i)T - 67iT^{2} \)
71 \( 1 + 2.92iT - 71T^{2} \)
73 \( 1 + (-2.51 - 2.51i)T + 73iT^{2} \)
79 \( 1 - 8.71iT - 79T^{2} \)
83 \( 1 + (-2.99 + 2.99i)T - 83iT^{2} \)
89 \( 1 + 9.72T + 89T^{2} \)
97 \( 1 + (-4.74 + 4.74i)T - 97iT^{2} \)
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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

−8.610905833843468011879839371667, −7.82236451143797545870380711029, −6.99173231641700821070476058214, −6.59887149628713875685291074480, −5.56368099952998296976804729293, −4.73952972846670617191514672312, −4.28430726169057202800501365329, −3.08089170336717241536480971263, −2.29985945266253156283004579131, −0.70406815817534533691427932371, 1.12870159742794260796439844728, 2.11454345679736615255568594777, 3.12464286353060354876683231621, 3.92232654573701573645484696858, 4.78675378995439360040761893574, 5.57499260095062393533049776711, 6.23228333437830771452084119651, 7.12241261811959331332349425814, 8.035648219806644287762528126600, 8.694130252224563941425830107783

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