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Properties

Label 2-630-63.38-c1-0-20
Degree $2$
Conductor $630$
Sign $0.246 - 0.969i$
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  + i·2-s + (1.53 + 0.809i)3-s − 4-s + (0.5 − 0.866i)5-s + (−0.809 + 1.53i)6-s + (2.54 + 0.721i)7-s − i·8-s + (1.69 + 2.47i)9-s + (0.866 + 0.5i)10-s + (0.559 − 0.322i)11-s + (−1.53 − 0.809i)12-s + (0.230 − 0.133i)13-s + (−0.721 + 2.54i)14-s + (1.46 − 0.921i)15-s + 16-s + (0.525 − 0.910i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.884 + 0.467i)3-s − 0.5·4-s + (0.223 − 0.387i)5-s + (−0.330 + 0.625i)6-s + (0.962 + 0.272i)7-s − 0.353i·8-s + (0.563 + 0.826i)9-s + (0.273 + 0.158i)10-s + (0.168 − 0.0973i)11-s + (−0.442 − 0.233i)12-s + (0.0639 − 0.0369i)13-s + (−0.192 + 0.680i)14-s + (0.378 − 0.237i)15-s + 0.250·16-s + (0.127 − 0.220i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.246 - 0.969i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.246 - 0.969i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.72216 + 1.33894i\)
\(L(\frac12)\) \(\approx\) \(1.72216 + 1.33894i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (-1.53 - 0.809i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-2.54 - 0.721i)T \)
good11 \( 1 + (-0.559 + 0.322i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.230 + 0.133i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.525 + 0.910i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.938 + 0.541i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.91 + 1.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.95 - 1.70i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 2.97iT - 31T^{2} \)
37 \( 1 + (-1.37 - 2.37i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.16 + 3.75i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (5.04 - 8.74i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.0537T + 47T^{2} \)
53 \( 1 + (10.1 + 5.86i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 5.70T + 59T^{2} \)
61 \( 1 + 3.54iT - 61T^{2} \)
67 \( 1 - 7.22T + 67T^{2} \)
71 \( 1 + 10.7iT - 71T^{2} \)
73 \( 1 + (13.9 + 8.03i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 4.32T + 79T^{2} \)
83 \( 1 + (-1.83 + 3.16i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (6.12 + 10.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-14.8 - 8.58i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.52891873021323187617267959166, −9.668228388659617172290457042562, −8.840582968952249279213424965922, −8.238285762735534205714293880830, −7.52104537145837990924451155367, −6.28802740142804606177040466621, −5.06678430932444429826611963185, −4.52566640667685125485336943558, −3.18754635812669111985367804237, −1.68556499442887278426271875119, 1.37534050661763788463730296072, 2.36760067943314469354604786466, 3.55886543703512298965152246132, 4.50694175229478532011474315947, 5.87586259262465622308618744691, 7.09523304722012626717732437865, 7.938635061939989774461246924839, 8.641995023554752218300530675438, 9.652031701844004051715815878076, 10.32626976900462939257089588042

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