Properties

Degree 2
Conductor $ 3^{2} \cdot 5 \cdot 7 $
Sign $0.426 + 0.904i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1.48 − 1.48i)2-s + 2.43i·4-s + (1.28 + 1.82i)5-s + (−1.97 − 1.75i)7-s + (0.640 − 0.640i)8-s + (0.798 − 4.63i)10-s + 2.67·11-s + (1.22 + 1.22i)13-s + (0.320 + 5.55i)14-s + 2.95·16-s + (4.74 − 4.74i)17-s + 6.01·19-s + (−4.43 + 3.13i)20-s + (−3.97 − 3.97i)22-s + (0.175 − 0.175i)23-s + ⋯
L(s)  = 1  + (−1.05 − 1.05i)2-s + 1.21i·4-s + (0.576 + 0.816i)5-s + (−0.746 − 0.665i)7-s + (0.226 − 0.226i)8-s + (0.252 − 1.46i)10-s + 0.805·11-s + (0.340 + 0.340i)13-s + (0.0857 + 1.48i)14-s + 0.738·16-s + (1.15 − 1.15i)17-s + 1.38·19-s + (−0.992 + 0.701i)20-s + (−0.847 − 0.847i)22-s + (0.0366 − 0.0366i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.426 + 0.904i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $0.426 + 0.904i$
motivic weight  =  \(1\)
character  :  $\chi_{315} (307, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 315,\ (\ :1/2),\ 0.426 + 0.904i)\)
\(L(1)\)  \(\approx\)  \(0.702547 - 0.445391i\)
\(L(\frac12)\)  \(\approx\)  \(0.702547 - 0.445391i\)
\(L(\frac{3}{2})\)   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 \( 1 + (-1.28 - 1.82i)T \)
7 \( 1 + (1.97 + 1.75i)T \)
good2 \( 1 + (1.48 + 1.48i)T + 2iT^{2} \)
11 \( 1 - 2.67T + 11T^{2} \)
13 \( 1 + (-1.22 - 1.22i)T + 13iT^{2} \)
17 \( 1 + (-4.74 + 4.74i)T - 17iT^{2} \)
19 \( 1 - 6.01T + 19T^{2} \)
23 \( 1 + (-0.175 + 0.175i)T - 23iT^{2} \)
29 \( 1 - 0.304iT - 29T^{2} \)
31 \( 1 + 7.25iT - 31T^{2} \)
37 \( 1 + (0.735 + 0.735i)T + 37iT^{2} \)
41 \( 1 - 7.05iT - 41T^{2} \)
43 \( 1 + (-0.304 + 0.304i)T - 43iT^{2} \)
47 \( 1 + (0.556 - 0.556i)T - 47iT^{2} \)
53 \( 1 + (-4.99 + 4.99i)T - 53iT^{2} \)
59 \( 1 - 7.98T + 59T^{2} \)
61 \( 1 - 5.53iT - 61T^{2} \)
67 \( 1 + (3.43 + 3.43i)T + 67iT^{2} \)
71 \( 1 + 15.3T + 71T^{2} \)
73 \( 1 + (-10.0 - 10.0i)T + 73iT^{2} \)
79 \( 1 + 11.2iT - 79T^{2} \)
83 \( 1 + (-4.88 - 4.88i)T + 83iT^{2} \)
89 \( 1 + 6.91T + 89T^{2} \)
97 \( 1 + (8.84 - 8.84i)T - 97iT^{2} \)
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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

−11.42584387701125491444598389606, −10.35621194575305713246741791723, −9.672174613261719868066303824256, −9.284611287590218686057283356341, −7.75631843342401345073617905338, −6.88011895183260327579331615672, −5.68778480823560344754367617259, −3.65951237575480396875889616968, −2.77034914889418226627487030400, −1.13514104450233459350373584402, 1.23796400683417233269945125099, 3.46973432164768618570026944954, 5.44535982475658814248495290997, 6.00175353306298689791845582755, 7.05867273473157312949459745044, 8.247376341430510854076821178071, 8.912471178258700101091117092678, 9.635786268449722229155011981244, 10.33526039988275178381864647681, 12.05971234128065436643923417627

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