Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.848 + 0.528i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 + 0.133i)2-s + (−1.5 + 0.866i)4-s + (−0.133 + 2.23i)5-s + (−2.5 − 0.866i)7-s + (1.36 − 1.36i)8-s + (−0.232 − 1.13i)10-s + (−1.36 − 2.36i)11-s + (−2 − 2i)13-s + (1.36 + 0.0980i)14-s + (1.23 − 2.13i)16-s + (−3.73 − i)17-s + (0.366 − 0.633i)19-s + (−1.73 − 3.46i)20-s + (1 + i)22-s + (−0.0358 − 0.133i)23-s + ⋯
L(s)  = 1  + (−0.353 + 0.0947i)2-s + (−0.750 + 0.433i)4-s + (−0.0599 + 0.998i)5-s + (−0.944 − 0.327i)7-s + (0.482 − 0.482i)8-s + (−0.0733 − 0.358i)10-s + (−0.411 − 0.713i)11-s + (−0.554 − 0.554i)13-s + (0.365 + 0.0262i)14-s + (0.308 − 0.533i)16-s + (−0.905 − 0.242i)17-s + (0.0839 − 0.145i)19-s + (−0.387 − 0.774i)20-s + (0.213 + 0.213i)22-s + (−0.00748 − 0.0279i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 + 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.848 + 0.528i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (82, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 315,\ (\ :1/2),\ -0.848 + 0.528i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{3,\;5,\;7\}$,\(F_p(T)\) is a polynomial of degree 2. If $p \in \{3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.133 - 2.23i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (0.5 - 0.133i)T + (1.73 - i)T^{2} \)
11 \( 1 + (1.36 + 2.36i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 + 2i)T + 13iT^{2} \)
17 \( 1 + (3.73 + i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.366 + 0.633i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.0358 + 0.133i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 3iT - 29T^{2} \)
31 \( 1 + (6.46 - 3.73i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (4.73 - 1.26i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 6.46iT - 41T^{2} \)
43 \( 1 + (-2.83 + 2.83i)T - 43iT^{2} \)
47 \( 1 + (2.36 + 8.83i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.83 + 1.83i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-4.09 - 7.09i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.33 - 0.767i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-2.86 + 10.6i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.26T + 71T^{2} \)
73 \( 1 + (3.46 - 12.9i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-2.83 - 1.63i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.09 - 2.09i)T + 83iT^{2} \)
89 \( 1 + (-0.330 + 0.571i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.92 - 5.92i)T - 97iT^{2} \)
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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

−11.01988965915660683459773698953, −10.28300927063859411199442970330, −9.452678417596889087818756325419, −8.458790976017573506282010299547, −7.38445168254937815534997134423, −6.66949696420315285969735052589, −5.27088811747544515284029233239, −3.77232174949928091027752476008, −2.89786278698498754613310790277, 0, 2.01241289316343666063431252314, 4.05757220432945931809831945093, 4.98297382150262257174462376505, 6.01342747302789142007375845135, 7.40934533161736134437569534928, 8.561105879653770955758932865974, 9.372860037563794455311362211296, 9.792981239999169455567177328440, 10.97889214881415987757610845828

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