Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $-0.0932 - 0.995i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.5 − 1.86i)2-s + (−1.5 + 0.866i)4-s + (−1.86 + 1.23i)5-s + (−2.5 − 0.866i)7-s + (−0.366 − 0.366i)8-s + (3.23 + 2.86i)10-s + (0.366 + 0.633i)11-s + (−2 + 2i)13-s + (−0.366 + 5.09i)14-s + (−2.23 + 3.86i)16-s + (−0.267 + i)17-s + (−1.36 + 2.36i)19-s + (1.73 − 3.46i)20-s + (1 − i)22-s + (−6.96 + 1.86i)23-s + ⋯
L(s)  = 1  + (−0.353 − 1.31i)2-s + (−0.750 + 0.433i)4-s + (−0.834 + 0.550i)5-s + (−0.944 − 0.327i)7-s + (−0.129 − 0.129i)8-s + (1.02 + 0.906i)10-s + (0.110 + 0.191i)11-s + (−0.554 + 0.554i)13-s + (−0.0978 + 1.36i)14-s + (−0.558 + 0.966i)16-s + (−0.0649 + 0.242i)17-s + (−0.313 + 0.542i)19-s + (0.387 − 0.774i)20-s + (0.213 − 0.213i)22-s + (−1.45 + 0.389i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0932 - 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0932 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.0932 - 0.995i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (208, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  1
Selberg data  =  $(2,\ 315,\ (\ :1/2),\ -0.0932 - 0.995i)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 \( 1 + (1.86 - 1.23i)T \)
7 \( 1 + (2.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 1.86i)T + (-1.73 + i)T^{2} \)
11 \( 1 + (-0.366 - 0.633i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2 - 2i)T - 13iT^{2} \)
17 \( 1 + (0.267 - i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (1.36 - 2.36i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.96 - 1.86i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 3iT - 29T^{2} \)
31 \( 1 + (-0.464 + 0.267i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.26 + 4.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.464iT - 41T^{2} \)
43 \( 1 + (5.83 + 5.83i)T + 43iT^{2} \)
47 \( 1 + (0.633 - 0.169i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.83 + 6.83i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (1.09 + 1.90i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (7.33 + 4.23i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.13 - 0.303i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.73T + 71T^{2} \)
73 \( 1 + (-3.46 - 0.928i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.83 + 3.36i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (3.09 - 3.09i)T - 83iT^{2} \)
89 \( 1 + (8.33 - 14.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-7.92 - 7.92i)T + 97iT^{2} \)
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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

−10.94355740486241854336244629898, −10.13934835345304747070563040400, −9.551254983366414389438120769195, −8.317863359401090406320972685449, −7.14077475868531075746984718603, −6.20659110733079697382415269340, −4.16652355826738446727253117151, −3.46410635088639982306731617573, −2.15281136732298662894809818789, 0, 3.00458943520830858995083971518, 4.56550392511836304380125359587, 5.70231369772050669472079267648, 6.65651615962366083596594509435, 7.57291525579443469611522539290, 8.424666384059848008544122160341, 9.152281283034785604741389051049, 10.18416604011106457992582606357, 11.62063202099331605020164063483

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