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Properties

Label 2-630-7.3-c2-0-7
Degree $2$
Conductor $630$
Sign $0.999 - 0.0242i$
Analytic cond. $17.1662$
Root an. cond. $4.14321$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 1.22i)2-s + (−0.999 + 1.73i)4-s + (−1.93 + 1.11i)5-s + (−6.38 + 2.86i)7-s + 2.82·8-s + (2.73 + 1.58i)10-s + (9.98 − 17.2i)11-s + 3.49i·13-s + (8.02 + 5.80i)14-s + (−2.00 − 3.46i)16-s + (−15.7 − 9.12i)17-s + (−21.3 + 12.3i)19-s − 4.47i·20-s − 28.2·22-s + (12.5 + 21.8i)23-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (−0.387 + 0.223i)5-s + (−0.912 + 0.408i)7-s + 0.353·8-s + (0.273 + 0.158i)10-s + (0.907 − 1.57i)11-s + 0.269i·13-s + (0.572 + 0.414i)14-s + (−0.125 − 0.216i)16-s + (−0.929 − 0.536i)17-s + (−1.12 + 0.647i)19-s − 0.223i·20-s − 1.28·22-s + (0.547 + 0.948i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0242i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 - 0.0242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0242i$
Analytic conductor: \(17.1662\)
Root analytic conductor: \(4.14321\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1),\ 0.999 - 0.0242i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.064224701\)
\(L(\frac12)\) \(\approx\) \(1.064224701\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
3 \( 1 \)
5 \( 1 + (1.93 - 1.11i)T \)
7 \( 1 + (6.38 - 2.86i)T \)
good11 \( 1 + (-9.98 + 17.2i)T + (-60.5 - 104. i)T^{2} \)
13 \( 1 - 3.49iT - 169T^{2} \)
17 \( 1 + (15.7 + 9.12i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (21.3 - 12.3i)T + (180.5 - 312. i)T^{2} \)
23 \( 1 + (-12.5 - 21.8i)T + (-264.5 + 458. i)T^{2} \)
29 \( 1 - 53.1T + 841T^{2} \)
31 \( 1 + (-26.0 - 15.0i)T + (480.5 + 832. i)T^{2} \)
37 \( 1 + (-23.3 - 40.5i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 31.5iT - 1.68e3T^{2} \)
43 \( 1 - 64.4T + 1.84e3T^{2} \)
47 \( 1 + (-24.3 + 14.0i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (32.4 - 56.1i)T + (-1.40e3 - 2.43e3i)T^{2} \)
59 \( 1 + (86.7 + 50.0i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-6.94 + 4.01i)T + (1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (8.13 - 14.0i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 107.T + 5.04e3T^{2} \)
73 \( 1 + (-44.7 - 25.8i)T + (2.66e3 + 4.61e3i)T^{2} \)
79 \( 1 + (-10.9 - 19.0i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 0.417iT - 6.88e3T^{2} \)
89 \( 1 + (-96.3 + 55.6i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 74.2iT - 9.40e3T^{2} \)
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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

−10.49984375601789625143676989222, −9.433375500694271978143582183381, −8.817760695870039218583846910143, −8.063820705286815007353869677731, −6.64180269572214771332388058543, −6.17195359466262481752770417191, −4.55091873395482705166550686769, −3.47813793355440867977765741879, −2.67671822813073338024405373648, −0.905244749959502596168461969977, 0.61779570629005299095606830577, 2.38928475193336061951260222612, 4.15740928136351258672512708662, 4.61440577990204580269058590478, 6.33871602796803835055849438342, 6.71108998824445348298785324517, 7.64438833232354224304848341466, 8.750378576399835789607666298018, 9.340150440357160361932752859805, 10.28553292160750372833719498247

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