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Properties

Label 2-630-63.47-c1-0-7
Degree $2$
Conductor $630$
Sign $0.743 - 0.668i$
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  + (−0.866 + 0.5i)2-s + (1.16 − 1.28i)3-s + (0.499 − 0.866i)4-s − 5-s + (−0.363 + 1.69i)6-s + (−1.57 + 2.12i)7-s + 0.999i·8-s + (−0.300 − 2.98i)9-s + (0.866 − 0.5i)10-s + 5.66i·11-s + (−0.531 − 1.64i)12-s + (2.43 − 1.40i)13-s + (0.300 − 2.62i)14-s + (−1.16 + 1.28i)15-s + (−0.5 − 0.866i)16-s + (3.51 + 6.08i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.670 − 0.741i)3-s + (0.249 − 0.433i)4-s − 0.447·5-s + (−0.148 + 0.691i)6-s + (−0.595 + 0.803i)7-s + 0.353i·8-s + (−0.100 − 0.994i)9-s + (0.273 − 0.158i)10-s + 1.70i·11-s + (−0.153 − 0.475i)12-s + (0.675 − 0.389i)13-s + (0.0802 − 0.702i)14-s + (−0.299 + 0.331i)15-s + (−0.125 − 0.216i)16-s + (0.851 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.743 - 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.743 - 0.668i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (551, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.743 - 0.668i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.12073 + 0.429891i\)
\(L(\frac12)\) \(\approx\) \(1.12073 + 0.429891i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-1.16 + 1.28i)T \)
5 \( 1 + T \)
7 \( 1 + (1.57 - 2.12i)T \)
good11 \( 1 - 5.66iT - 11T^{2} \)
13 \( 1 + (-2.43 + 1.40i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.51 - 6.08i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.76 - 2.17i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 4.07iT - 23T^{2} \)
29 \( 1 + (-5.28 - 3.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.319 - 0.184i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.783 + 1.35i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.39 - 2.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.18 + 5.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.59 + 4.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.02 - 1.74i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.32 - 9.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-12.6 + 7.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.36 - 2.36i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + (14.3 - 8.29i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.37 - 7.57i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.34 - 14.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.332 - 0.575i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-1.68 - 0.971i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.31889089992832998845130966216, −9.750080334056334594631445935780, −8.700556739698713439688346191971, −8.163987677745565236541973337188, −7.27044813306897666805785246889, −6.49522295032316614631222914020, −5.53860649020315935486030198094, −3.92036572661949507405223646244, −2.69553675201294553639723591588, −1.42062411935293469990971564488, 0.840684990308132201785092990972, 3.09507656580674923487906688515, 3.37479593062257064099984001055, 4.65247364642649624939577683994, 6.02910099153138745460115116424, 7.33990186950095305224494380058, 7.988506338805154033717915213479, 8.954890372458604895601602279793, 9.557985663384092680984853233070, 10.38162896763129200903481273892

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