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Properties

Label 2-630-63.5-c1-0-14
Degree $2$
Conductor $630$
Sign $-0.109 + 0.993i$
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 + (−1.61 − 0.629i)3-s − 4-s + (0.5 + 0.866i)5-s + (−0.629 + 1.61i)6-s + (−0.166 − 2.64i)7-s + i·8-s + (2.20 + 2.03i)9-s + (0.866 − 0.5i)10-s + (4.91 + 2.83i)11-s + (1.61 + 0.629i)12-s + (1.57 + 0.909i)13-s + (−2.64 + 0.166i)14-s + (−0.261 − 1.71i)15-s + 16-s + (−2.85 − 4.95i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.931 − 0.363i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.257 + 0.658i)6-s + (−0.0629 − 0.998i)7-s + 0.353i·8-s + (0.735 + 0.677i)9-s + (0.273 − 0.158i)10-s + (1.48 + 0.856i)11-s + (0.465 + 0.181i)12-s + (0.436 + 0.252i)13-s + (−0.705 + 0.0445i)14-s + (−0.0674 − 0.442i)15-s + 0.250·16-s + (−0.693 − 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.109 + 0.993i$
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: \((2,\ 630,\ (\ :1/2),\ -0.109 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.752609 - 0.840337i\)
\(L(\frac12)\) \(\approx\) \(0.752609 - 0.840337i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 + 0.629i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.166 + 2.64i)T \)
good11 \( 1 + (-4.91 - 2.83i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.57 - 0.909i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (2.85 + 4.95i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.07 - 0.619i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.01 + 1.74i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.25 - 1.30i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + 8.91iT - 31T^{2} \)
37 \( 1 + (2.02 - 3.51i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.21 + 10.7i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (2.45 + 4.26i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + (-0.844 + 0.487i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 7.44iT - 61T^{2} \)
67 \( 1 + 0.0689T + 67T^{2} \)
71 \( 1 + 6.62iT - 71T^{2} \)
73 \( 1 + (7.47 - 4.31i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 1.36T + 79T^{2} \)
83 \( 1 + (6.04 + 10.4i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.88 - 13.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1.67 - 0.969i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.51467405420965285296371234605, −9.742210624223209195093293087891, −8.937277198563149617797414861517, −7.24890135126755286863766815244, −6.99660021076529941050347052507, −5.84352801628899786617353556798, −4.56231830697537878672611345736, −3.87429215820439915844724465163, −2.13021453837268922978606595667, −0.858005443467726613383486680440, 1.27663803018748942287787849097, 3.51195209390934787049443467773, 4.55622048818653788424313884420, 5.67319658749945449849854444413, 6.10581194643327861834610901954, 6.93865676944283328990952838142, 8.495586310453062895958965816255, 8.953632000674928295099788086197, 9.764049837087717408682158856325, 10.96438456577953534430404540307

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