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Properties

Label 2-630-105.23-c1-0-12
Degree $2$
Conductor $630$
Sign $-0.531 + 0.847i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (−1.56 − 1.59i)5-s + (1.65 − 2.06i)7-s + (−0.707 + 0.707i)8-s + (1.92 + 1.13i)10-s + (−0.565 + 0.326i)11-s + (0.771 + 0.771i)13-s + (−1.06 + 2.42i)14-s + (0.500 − 0.866i)16-s + (0.554 − 2.06i)17-s + (−4.34 − 2.50i)19-s + (−2.15 − 0.596i)20-s + (0.461 − 0.461i)22-s + (−0.242 − 0.905i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (−0.701 − 0.712i)5-s + (0.626 − 0.779i)7-s + (−0.249 + 0.249i)8-s + (0.609 + 0.358i)10-s + (−0.170 + 0.0984i)11-s + (0.214 + 0.214i)13-s + (−0.285 + 0.647i)14-s + (0.125 − 0.216i)16-s + (0.134 − 0.501i)17-s + (−0.996 − 0.575i)19-s + (−0.481 − 0.133i)20-s + (0.0984 − 0.0984i)22-s + (−0.0505 − 0.188i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.531 + 0.847i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.531 + 0.847i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.531 + 0.847i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.312122 - 0.563977i\)
\(L(\frac12)\) \(\approx\) \(0.312122 - 0.563977i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (1.56 + 1.59i)T \)
7 \( 1 + (-1.65 + 2.06i)T \)
good11 \( 1 + (0.565 - 0.326i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.771 - 0.771i)T + 13iT^{2} \)
17 \( 1 + (-0.554 + 2.06i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (4.34 + 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.242 + 0.905i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 9.79T + 29T^{2} \)
31 \( 1 + (0.897 + 1.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.67 + 6.26i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 7.89iT - 41T^{2} \)
43 \( 1 + (0.0657 + 0.0657i)T + 43iT^{2} \)
47 \( 1 + (-3.18 + 0.854i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-7.13 - 1.91i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (3.39 + 5.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.33 - 7.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6.21 + 1.66i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 - 7.68iT - 71T^{2} \)
73 \( 1 + (-2.58 + 9.62i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (10.3 + 5.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.838 - 0.838i)T - 83iT^{2} \)
89 \( 1 + (-8.65 + 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.59 - 8.59i)T - 97iT^{2} \)
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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

−10.38318465120983982694725238970, −9.153539988970876124648427839748, −8.659529845652887488562308953096, −7.55485447879232585209945439514, −7.22130419699162985820090905670, −5.76508122906009851907638570149, −4.68125622176759245376019203597, −3.77003427039673793917056895653, −1.93596090602784101419933346484, −0.43292100266965811093443244140, 1.81169050558988418867123563572, 3.04174586077199386214951049433, 4.15957506124801541098945480181, 5.60632010603751757133391576846, 6.54318949661428176525627738857, 7.67592482826025874330795337035, 8.213575398417859704750051099491, 9.037121446449227772146637159721, 10.16558757717430595974824405786, 10.89457534223765739976271405224

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