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Properties

Label 2-630-315.164-c1-0-4
Degree $2$
Conductor $630$
Sign $0.652 - 0.757i$
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  − 2-s + (−0.241 − 1.71i)3-s + 4-s + (−1.71 + 1.43i)5-s + (0.241 + 1.71i)6-s + (−2.63 − 0.178i)7-s − 8-s + (−2.88 + 0.829i)9-s + (1.71 − 1.43i)10-s + (3.12 − 1.80i)11-s + (−0.241 − 1.71i)12-s + (−1.68 − 2.92i)13-s + (2.63 + 0.178i)14-s + (2.87 + 2.59i)15-s + 16-s + (6.40 + 3.69i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.139 − 0.990i)3-s + 0.5·4-s + (−0.767 + 0.640i)5-s + (0.0987 + 0.700i)6-s + (−0.997 − 0.0674i)7-s − 0.353·8-s + (−0.960 + 0.276i)9-s + (0.543 − 0.452i)10-s + (0.942 − 0.544i)11-s + (−0.0698 − 0.495i)12-s + (−0.467 − 0.810i)13-s + (0.705 + 0.0477i)14-s + (0.741 + 0.670i)15-s + 0.250·16-s + (1.55 + 0.897i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.652 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.652 - 0.757i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (479, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.652 - 0.757i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.493440 + 0.226344i\)
\(L(\frac12)\) \(\approx\) \(0.493440 + 0.226344i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (0.241 + 1.71i)T \)
5 \( 1 + (1.71 - 1.43i)T \)
7 \( 1 + (2.63 + 0.178i)T \)
good11 \( 1 + (-3.12 + 1.80i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (1.68 + 2.92i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-6.40 - 3.69i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (4.15 - 2.39i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.99 - 6.91i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-7.66 - 4.42i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 6.44iT - 31T^{2} \)
37 \( 1 + (-5.56 + 3.21i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (1.31 + 2.27i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.660 - 0.381i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 8.22iT - 47T^{2} \)
53 \( 1 + (1.47 - 2.55i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.12T + 59T^{2} \)
61 \( 1 + 1.68iT - 61T^{2} \)
67 \( 1 - 4.74iT - 67T^{2} \)
71 \( 1 - 1.40iT - 71T^{2} \)
73 \( 1 + (2.78 - 4.81i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 0.172T + 79T^{2} \)
83 \( 1 + (11.5 + 6.65i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.08 + 12.2i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.18 - 7.25i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.59866245443598352704114551445, −10.01327752985337351204289888919, −8.749669089718408645838520198423, −7.970863187635496579366997307003, −7.30992880540500811654781130524, −6.38243006322573700235385554314, −5.80520170258426923499235500322, −3.66826721829906069464871453042, −2.91616111102129862221775501392, −1.18360978484326107408590199919, 0.43803319679559634358735856574, 2.67741722685131610963987438260, 3.98110125780494469188987378164, 4.67192721006958502037314750949, 6.09648527522764293552993244104, 6.92249291245940573127818631402, 8.133211943640864538441353380013, 8.903287633180130519949743747996, 9.763370899596405601391262254403, 10.01159885793239549679935555300

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