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Properties

Label 2-630-63.38-c1-0-19
Degree $2$
Conductor $630$
Sign $0.967 + 0.254i$
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 + (0.361 − 1.69i)3-s − 4-s + (−0.5 + 0.866i)5-s + (1.69 + 0.361i)6-s + (−1.96 + 1.77i)7-s − i·8-s + (−2.73 − 1.22i)9-s + (−0.866 − 0.5i)10-s + (3.53 − 2.03i)11-s + (−0.361 + 1.69i)12-s + (5.20 − 3.00i)13-s + (−1.77 − 1.96i)14-s + (1.28 + 1.16i)15-s + 16-s + (0.641 − 1.11i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (0.208 − 0.977i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (0.691 + 0.147i)6-s + (−0.741 + 0.670i)7-s − 0.353i·8-s + (−0.912 − 0.408i)9-s + (−0.273 − 0.158i)10-s + (1.06 − 0.614i)11-s + (−0.104 + 0.488i)12-s + (1.44 − 0.832i)13-s + (−0.474 − 0.524i)14-s + (0.332 + 0.299i)15-s + 0.250·16-s + (0.155 − 0.269i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 + 0.254i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.967 + 0.254i$
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: \((2,\ 630,\ (\ :1/2),\ 0.967 + 0.254i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40891 - 0.182146i\)
\(L(\frac12)\) \(\approx\) \(1.40891 - 0.182146i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.361 + 1.69i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.96 - 1.77i)T \)
good11 \( 1 + (-3.53 + 2.03i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.20 + 3.00i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-0.641 + 1.11i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.90 + 1.67i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.89 - 1.09i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-6.21 - 3.59i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.84iT - 31T^{2} \)
37 \( 1 + (2.90 + 5.02i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.36 + 2.37i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2.43 - 4.22i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 8.27T + 47T^{2} \)
53 \( 1 + (-7.48 - 4.32i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 1.26T + 59T^{2} \)
61 \( 1 - 2.96iT - 61T^{2} \)
67 \( 1 + 8.62T + 67T^{2} \)
71 \( 1 + 14.1iT - 71T^{2} \)
73 \( 1 + (6.02 + 3.47i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 12.1T + 79T^{2} \)
83 \( 1 + (6.87 - 11.9i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-7.29 - 12.6i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-16.8 - 9.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.60496925033683535760972489616, −9.193439741251773652341267753345, −8.790570316621497570495458622368, −7.83267599667853567760713748489, −6.91267840312675437428730490906, −6.16837975410705556492322697365, −5.59581942256145408095873076422, −3.71064145504960079753115574761, −2.90003688146609150232753874324, −0.935831349333574292499023688978, 1.33370893733753415369064887511, 3.20553761567222272014010139567, 3.94907837928346540675558513370, 4.62255316259898019616158813032, 6.00096999841704509644437612565, 7.05279897747081347501280215747, 8.574375870585221954772387399395, 8.931656283832013941682170994772, 9.984715776937499260283514229813, 10.38460572213539340833951716926

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