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Properties

Label 2-630-63.38-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.936 - 0.349i$
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 + (−0.117 − 1.72i)3-s − 4-s + (0.5 − 0.866i)5-s + (1.72 − 0.117i)6-s + (−2.63 + 0.206i)7-s − i·8-s + (−2.97 + 0.405i)9-s + (0.866 + 0.5i)10-s + (−0.924 + 0.533i)11-s + (0.117 + 1.72i)12-s + (−3.43 + 1.98i)13-s + (−0.206 − 2.63i)14-s + (−1.55 − 0.762i)15-s + 16-s + (−3.26 + 5.65i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.0677 − 0.997i)3-s − 0.5·4-s + (0.223 − 0.387i)5-s + (0.705 − 0.0478i)6-s + (−0.996 + 0.0781i)7-s − 0.353i·8-s + (−0.990 + 0.135i)9-s + (0.273 + 0.158i)10-s + (−0.278 + 0.160i)11-s + (0.0338 + 0.498i)12-s + (−0.952 + 0.549i)13-s + (−0.0552 − 0.704i)14-s + (−0.401 − 0.196i)15-s + 0.250·16-s + (−0.792 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.936 - 0.349i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0368503 + 0.203942i\)
\(L(\frac12)\) \(\approx\) \(0.0368503 + 0.203942i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.117 + 1.72i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.63 - 0.206i)T \)
good11 \( 1 + (0.924 - 0.533i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.43 - 1.98i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.26 - 5.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.00 + 0.582i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-6.38 - 3.68i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (5.89 + 3.40i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.17iT - 31T^{2} \)
37 \( 1 + (-2.89 - 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (5.22 + 9.05i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.665 + 1.15i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 + (7.55 + 4.36i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.26T + 59T^{2} \)
61 \( 1 + 3.49iT - 61T^{2} \)
67 \( 1 - 5.96T + 67T^{2} \)
71 \( 1 + 0.707iT - 71T^{2} \)
73 \( 1 + (6.25 + 3.60i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 6.87T + 79T^{2} \)
83 \( 1 + (2.68 - 4.65i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-2.09 - 3.62i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (4.59 + 2.65i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.03990574863888385921960871827, −9.821355022768096131702754873521, −9.098362856734989101605877199619, −8.262093532661926110615188545679, −7.23086389683040738114961357542, −6.64940613691242146903137603027, −5.78070621317842930473976582092, −4.81324101714158654109400925576, −3.29465105719210579810440072410, −1.83573094249315972702146199949, 0.10687805702704846641758293010, 2.70326414765048453435599865727, 3.18241607117369472747326874558, 4.54987271066330046094814784681, 5.34170224794239086052704095943, 6.50409500178601604562132946486, 7.61874420937052212894644404113, 9.000770902563354760073836229506, 9.533259345384468984822724562711, 10.15675781610341665872393787963

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