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Properties

Label 2-630-105.2-c1-0-4
Degree $2$
Conductor $630$
Sign $0.885 + 0.465i$
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  + (−0.258 − 0.965i)2-s + (−0.866 + 0.499i)4-s + (−2.16 − 0.561i)5-s + (2.06 + 1.65i)7-s + (0.707 + 0.707i)8-s + (0.0180 + 2.23i)10-s + (−0.565 + 0.326i)11-s + (0.771 − 0.771i)13-s + (1.06 − 2.42i)14-s + (0.500 − 0.866i)16-s + (2.06 + 0.554i)17-s + (4.34 + 2.50i)19-s + (2.15 − 0.596i)20-s + (0.461 + 0.461i)22-s + (−0.905 + 0.242i)23-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.433 + 0.249i)4-s + (−0.967 − 0.251i)5-s + (0.779 + 0.626i)7-s + (0.249 + 0.249i)8-s + (0.00570 + 0.707i)10-s + (−0.170 + 0.0984i)11-s + (0.214 − 0.214i)13-s + (0.285 − 0.647i)14-s + (0.125 − 0.216i)16-s + (0.501 + 0.134i)17-s + (0.996 + 0.575i)19-s + (0.481 − 0.133i)20-s + (0.0984 + 0.0984i)22-s + (−0.188 + 0.0505i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.885 + 0.465i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.885 + 0.465i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.885 + 0.465i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.16701 - 0.287919i\)
\(L(\frac12)\) \(\approx\) \(1.16701 - 0.287919i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 + 0.965i)T \)
3 \( 1 \)
5 \( 1 + (2.16 + 0.561i)T \)
7 \( 1 + (-2.06 - 1.65i)T \)
good11 \( 1 + (0.565 - 0.326i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.771 + 0.771i)T - 13iT^{2} \)
17 \( 1 + (-2.06 - 0.554i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-4.34 - 2.50i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.905 - 0.242i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 9.79T + 29T^{2} \)
31 \( 1 + (0.897 + 1.55i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.26 + 1.67i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 7.89iT - 41T^{2} \)
43 \( 1 + (0.0657 - 0.0657i)T - 43iT^{2} \)
47 \( 1 + (-0.854 - 3.18i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-1.91 + 7.13i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.39 - 5.87i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.33 - 7.51i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.66 + 6.21i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 7.68iT - 71T^{2} \)
73 \( 1 + (9.62 + 2.58i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-10.3 - 5.96i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.838 - 0.838i)T + 83iT^{2} \)
89 \( 1 + (8.65 - 14.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (8.59 + 8.59i)T + 97iT^{2} \)
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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

−10.66017117126564361148316939486, −9.734463871475498076869528985157, −8.677825968174482097992247481955, −8.104127116064113819742100867453, −7.33748468745295242366265810030, −5.75600869731189895435859652484, −4.82104243005425939055449501548, −3.82492780487037742069879783212, −2.67455230398984358539847890236, −1.12094489990799312120776082896, 0.965563737984618785498905769185, 3.09180362644849884143213102116, 4.32585761807625905052880684379, 5.03860894393192451520828367976, 6.38712020550175277504515590443, 7.29323597279867724208809441223, 7.920529582865560418369368190562, 8.606789276781935597445952583832, 9.778252581403117598154321903924, 10.66384202965047857466813741026

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