Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 $
Sign $0.894 - 0.447i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  i·2-s − 4-s + i·7-s + i·8-s + i·13-s + 14-s + 16-s − 3i·17-s − 2·19-s + 3i·23-s + 26-s i·28-s + 3·29-s − 31-s i·32-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s + 0.377i·7-s + 0.353i·8-s + 0.277i·13-s + 0.267·14-s + 0.250·16-s − 0.727i·17-s − 0.458·19-s + 0.625i·23-s + 0.196·26-s − 0.188i·28-s + 0.557·29-s − 0.179·31-s − 0.176i·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(3150\)    =    \(2 \cdot 3^{2} \cdot 5^{2} \cdot 7\)
\( \varepsilon \)  =  $0.894 - 0.447i$
motivic weight  =  \(1\)
character  :  $\chi_{3150} (2899, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(2,\ 3150,\ (\ :1/2),\ 0.894 - 0.447i)$
$L(1)$  $\approx$  $1.333052697$
$L(\frac12)$  $\approx$  $1.333052697$
$L(\frac{3}{2})$   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 \( 1 + iT \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - iT \)
good11 \( 1 + 11T^{2} \)
13 \( 1 - iT - 13T^{2} \)
17 \( 1 + 3iT - 17T^{2} \)
19 \( 1 + 2T + 19T^{2} \)
23 \( 1 - 3iT - 23T^{2} \)
29 \( 1 - 3T + 29T^{2} \)
31 \( 1 + T + 31T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 - 3T + 41T^{2} \)
43 \( 1 - 7iT - 43T^{2} \)
47 \( 1 + 6iT - 47T^{2} \)
53 \( 1 - 9iT - 53T^{2} \)
59 \( 1 - 3T + 59T^{2} \)
61 \( 1 + T + 61T^{2} \)
67 \( 1 - 8iT - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 - 15iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−8.892873197570309396618935235142, −8.143699875383429433307235246115, −7.31353699695357714789958586425, −6.43572334442883342116061027384, −5.56546816637444679235070848430, −4.78443066126930797220887524463, −3.97704346989484432646435547383, −2.99458037291442977361376496155, −2.22236095699846338261164221695, −1.06717930086113082484273531165, 0.47440218472797002822762346911, 1.88979011876853662396838087607, 3.16558000415867748122058097079, 4.09276954009409360383077629486, 4.78183526035563087960705749112, 5.73699273207765516982578236559, 6.38263968118976671161944492532, 7.11688400372109128256356569296, 7.86850025021864003922919082565, 8.514491285592630417106541837167

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