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Properties

Label 2-630-105.23-c1-0-9
Degree $2$
Conductor $630$
Sign $0.975 + 0.218i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 + 0.218i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.975 + 0.218i$
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: \((2,\ 630,\ (\ :1/2),\ 0.975 + 0.218i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.62114 - 0.289511i\)
\(L(\frac12)\) \(\approx\) \(2.62114 - 0.289511i\)
\(L(\frac{3}{2})\) 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

−10.72626650055570246592952982052, −10.02412790340491893901880031269, −8.874976943552622086829709115809, −7.75892081616201607112304787009, −6.72841062278592812263391546724, −6.15580199323765506979415601798, −4.90574132886672735421174980688, −4.01888854319458043994284704974, −2.78468106497424021631484167385, −1.56012728168678279119282904123, 1.63189488757420798688198352593, 2.78351331472165227469148392847, 4.36261736762620352014206612987, 5.10208966885313968208780200542, 5.95080224138586346589050397325, 6.78853586052621366665295566976, 8.241766649761066058069098179421, 8.634174036100195102355208094208, 9.768601442543050068854342706550, 10.68692430517899457176449161007

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