Properties

Degree 2
Conductor $ 2 \cdot 3^{2} \cdot 5 \cdot 7 $
Sign $-0.376 - 0.926i$
Motivic weight 1
Primitive yes
Self-dual no
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 − 0.5i)2-s + (1.65 + 0.521i)3-s + (0.499 + 0.866i)4-s − 5-s + (−1.16 − 1.27i)6-s + (−2.02 + 1.69i)7-s − 0.999i·8-s + (2.45 + 1.72i)9-s + (0.866 + 0.5i)10-s + 5.51i·11-s + (0.373 + 1.69i)12-s + (−5.67 − 3.27i)13-s + (2.60 − 0.455i)14-s + (−1.65 − 0.521i)15-s + (−0.5 + 0.866i)16-s + (1.26 − 2.19i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.953 + 0.301i)3-s + (0.249 + 0.433i)4-s − 0.447·5-s + (−0.477 − 0.521i)6-s + (−0.767 + 0.641i)7-s − 0.353i·8-s + (0.818 + 0.574i)9-s + (0.273 + 0.158i)10-s + 1.66i·11-s + (0.107 + 0.488i)12-s + (−1.57 − 0.908i)13-s + (0.696 − 0.121i)14-s + (−0.426 − 0.134i)15-s + (−0.125 + 0.216i)16-s + (0.307 − 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.376 - 0.926i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
\( \varepsilon \)  =  $-0.376 - 0.926i$
motivic weight  =  \(1\)
character  :  $\chi_{630} (311, \cdot )$
primitive  :  yes
self-dual  :  no
analytic rank  =  \(0\)
Selberg data  =  \((2,\ 630,\ (\ :1/2),\ -0.376 - 0.926i)\)
\(L(1)\)  \(\approx\)  \(0.466965 + 0.693585i\)
\(L(\frac12)\)  \(\approx\)  \(0.466965 + 0.693585i\)
\(L(\frac{3}{2})\)   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 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (-1.65 - 0.521i)T \)
5 \( 1 + T \)
7 \( 1 + (2.02 - 1.69i)T \)
good11 \( 1 - 5.51iT - 11T^{2} \)
13 \( 1 + (5.67 + 3.27i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.26 + 2.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (6.11 - 3.53i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.21iT - 23T^{2} \)
29 \( 1 + (-6.29 + 3.63i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (5.33 - 3.08i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.00 + 1.74i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.97 - 5.15i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.74 - 4.76i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.69 + 4.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.56 - 1.48i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.95 + 3.39i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.512 - 0.295i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-1.48 - 2.57i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + (-9.76 - 5.63i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.619 - 1.07i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.32 + 7.48i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-7.84 - 13.5i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.197 + 0.113i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.35183550744071633787701463924, −9.928027669610607813555916212451, −9.334459859643806828446622640304, −8.298037000133737204133398786910, −7.55816905202210010791814733704, −6.84783502435222980591215121209, −5.16189688824992628818191566883, −4.09083199136771410286565403482, −2.89658551040290877169161282500, −2.10740924404406150214238845295, 0.46559417303241220102925705007, 2.34485073109531480811237648370, 3.48004794283572645107862294581, 4.57854513701396084755316416555, 6.28246683727010675681497087358, 6.91839758274407341349457683402, 7.72476566310466219238017251452, 8.696568693137226518536848941829, 9.113750116775634169472514741751, 10.24598760345433029849936774456

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