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Properties

Label 2-630-105.2-c1-0-14
Degree $2$
Conductor $630$
Sign $-0.909 - 0.416i$
Analytic cond. $5.03057$
Root an. cond. $2.24289$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−1.99 − 1.01i)5-s + (1.78 − 1.94i)7-s + (0.707 + 0.707i)8-s + (−0.469 + 2.18i)10-s + (−2.64 + 1.52i)11-s + (1 − i)13-s + (−2.34 − 1.22i)14-s + (0.500 − 0.866i)16-s + (−4.16 − 1.11i)17-s + (−5.47 − 3.15i)19-s + (2.23 − 0.111i)20-s + (2.15 + 2.15i)22-s + (1.62 − 0.435i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.889 − 0.456i)5-s + (0.676 − 0.736i)7-s + (0.249 + 0.249i)8-s + (−0.148 + 0.691i)10-s + (−0.797 + 0.460i)11-s + (0.277 − 0.277i)13-s + (−0.626 − 0.327i)14-s + (0.125 − 0.216i)16-s + (−1.01 − 0.270i)17-s + (−1.25 − 0.724i)19-s + (0.499 − 0.0250i)20-s + (0.460 + 0.460i)22-s + (0.339 − 0.0908i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.909 - 0.416i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.909 - 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0856642 + 0.392542i\)
\(L(\frac12)\) \(\approx\) \(0.0856642 + 0.392542i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (1.99 + 1.01i)T \)
7 \( 1 + (-1.78 + 1.94i)T \)
good11 \( 1 + (2.64 - 1.52i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (4.16 + 1.11i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.47 + 3.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.62 + 0.435i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 5.88T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.83 - 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 2.82iT - 41T^{2} \)
43 \( 1 + (7.63 - 7.63i)T - 43iT^{2} \)
47 \( 1 + (-0.0819 - 0.305i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (0.422 - 1.57i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (5.99 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 - 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.89 + 14.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 1.86iT - 71T^{2} \)
73 \( 1 + (-3.66 - 0.982i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.14 + 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-8.67 - 8.67i)T + 83iT^{2} \)
89 \( 1 + (-6.32 + 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.84 + 5.84i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.41144466902882300091878984897, −9.159674024885000763255438006609, −8.397609768217314991717131906199, −7.68302101573369338584077102195, −6.75935183904477555753957403537, −4.96894191830702078834327657514, −4.50798092213506967403294564665, −3.35752893487138527630001109385, −1.87788707864263767593109403774, −0.22532130991183929534283103240, 2.15267597897024642496982389343, 3.68647843519153714978528760617, 4.72434094843471744512365076903, 5.77368000619226405158178815308, 6.68998087406009370350560239954, 7.68617794390850099965932941026, 8.427045976535743613672986323384, 8.922952931988222910027754044288, 10.37615946657777611519262915652, 11.00035771250915887032989272869

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