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Properties

Label 2-630-63.5-c1-0-27
Degree $2$
Conductor $630$
Sign $-0.994 - 0.104i$
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 + (0.352 − 1.69i)3-s − 4-s + (−0.5 − 0.866i)5-s + (−1.69 − 0.352i)6-s + (2.55 + 0.696i)7-s + i·8-s + (−2.75 − 1.19i)9-s + (−0.866 + 0.5i)10-s + (1.26 + 0.732i)11-s + (−0.352 + 1.69i)12-s + (−6.03 − 3.48i)13-s + (0.696 − 2.55i)14-s + (−1.64 + 0.542i)15-s + 16-s + (−3.30 − 5.72i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.203 − 0.979i)3-s − 0.5·4-s + (−0.223 − 0.387i)5-s + (−0.692 − 0.143i)6-s + (0.964 + 0.263i)7-s + 0.353i·8-s + (−0.917 − 0.398i)9-s + (−0.273 + 0.158i)10-s + (0.382 + 0.220i)11-s + (−0.101 + 0.489i)12-s + (−1.67 − 0.966i)13-s + (0.186 − 0.682i)14-s + (−0.424 + 0.140i)15-s + 0.250·16-s + (−0.801 − 1.38i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.104i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.994 - 0.104i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.994 - 0.104i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0608679 + 1.16064i\)
\(L(\frac12)\) \(\approx\) \(0.0608679 + 1.16064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.352 + 1.69i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.55 - 0.696i)T \)
good11 \( 1 + (-1.26 - 0.732i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (6.03 + 3.48i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (3.30 + 5.72i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.08 + 1.78i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-5.51 + 3.18i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.49 + 0.862i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.84iT - 31T^{2} \)
37 \( 1 + (2.75 - 4.77i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-0.632 + 1.09i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.24 + 5.62i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.39T + 47T^{2} \)
53 \( 1 + (-5.60 + 3.23i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 1.49T + 59T^{2} \)
61 \( 1 - 3.73iT - 61T^{2} \)
67 \( 1 - 12.5T + 67T^{2} \)
71 \( 1 + 14.2iT - 71T^{2} \)
73 \( 1 + (1.11 - 0.640i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.994T + 79T^{2} \)
83 \( 1 + (5.93 + 10.2i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-2.18 + 3.78i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-9.04 + 5.22i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.27419606689091581218315410415, −9.023336405854107015970116071999, −8.593518828773221686274291561771, −7.51777113908411704045570938801, −6.84552295608655076110764177471, −5.18905403888987464604196764199, −4.72492140164767055055349627020, −2.93852420686230266514052436445, −2.10624579523580455646000511360, −0.60849635856559628993597423979, 2.23980205277307343445599875490, 3.92894745379121573123718363131, 4.47895071592582540077019383532, 5.47437228583518390937145945939, 6.64736931192081203691661260604, 7.58571070905907137732043165004, 8.442650770414374623491027370400, 9.207880009623218811599645878730, 10.08407266176172731511915564899, 10.98096477574242540065023020370

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