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Properties

Label 2-630-9.7-c1-0-12
Degree $2$
Conductor $630$
Sign $0.974 + 0.226i$
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  + (−0.5 + 0.866i)2-s + (−1.25 + 1.19i)3-s + (−0.499 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.403 − 1.68i)6-s + (−0.5 + 0.866i)7-s + 0.999·8-s + (0.160 − 2.99i)9-s − 0.999·10-s + (2.91 − 5.05i)11-s + (1.66 + 0.492i)12-s + (−2.41 − 4.18i)13-s + (−0.499 − 0.866i)14-s + (−1.66 − 0.492i)15-s + (−0.5 + 0.866i)16-s − 4.70·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.725 + 0.687i)3-s + (−0.249 − 0.433i)4-s + (0.223 + 0.387i)5-s + (−0.164 − 0.687i)6-s + (−0.188 + 0.327i)7-s + 0.353·8-s + (0.0534 − 0.998i)9-s − 0.316·10-s + (0.879 − 1.52i)11-s + (0.479 + 0.142i)12-s + (−0.670 − 1.16i)13-s + (−0.133 − 0.231i)14-s + (−0.428 − 0.127i)15-s + (−0.125 + 0.216i)16-s − 1.14·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.974 + 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.974 + 0.226i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (421, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.974 + 0.226i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.733504 - 0.0839982i\)
\(L(\frac12)\) \(\approx\) \(0.733504 - 0.0839982i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (1.25 - 1.19i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good11 \( 1 + (-2.91 + 5.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.41 + 4.18i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + 4.70T + 17T^{2} \)
19 \( 1 - 0.707T + 19T^{2} \)
23 \( 1 + (1.41 + 2.45i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.41 + 7.65i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 + (-0.0824 - 0.142i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.12 + 7.14i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + 10.5T + 53T^{2} \)
59 \( 1 + (3.48 + 6.02i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.56 - 2.70i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.18 - 12.4i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.5T + 71T^{2} \)
73 \( 1 + 4.41T + 73T^{2} \)
79 \( 1 + (-0.707 + 1.22i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.12 + 12.3i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 9.41T + 89T^{2} \)
97 \( 1 + (-6.33 + 10.9i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.47114076192125499921070464431, −9.715524571862491905553915590326, −8.884964160368869999934482972531, −8.056567375599730580216476961187, −6.63019073516528830135589046898, −6.16411166853608509036761109889, −5.31448930947522462486530065926, −4.16117980172340582968552107102, −2.87708131283625791577493196856, −0.53530645338903379925161210169, 1.38479270518585374596424950258, 2.30470178997068449334091354006, 4.27787571569076114443279907230, 4.83464531789854644802397747510, 6.42905726835322619473480990407, 6.99716127978663913743900745244, 7.911241298989479954923088045501, 9.266392690945653580745645453408, 9.628790175518042089641738268080, 10.76763087963324912398743885071

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