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Properties

Label 2-630-315.214-c1-0-29
Degree $2$
Conductor $630$
Sign $0.802 + 0.596i$
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 + (1.57 + 0.719i)3-s − 4-s + (0.173 − 2.22i)5-s + (0.719 − 1.57i)6-s + (2.30 + 1.29i)7-s + i·8-s + (1.96 + 2.26i)9-s + (−2.22 − 0.173i)10-s + (1.72 + 2.98i)11-s + (−1.57 − 0.719i)12-s + (−0.648 + 0.374i)13-s + (1.29 − 2.30i)14-s + (1.87 − 3.38i)15-s + 16-s + (4.33 + 2.50i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.909 + 0.415i)3-s − 0.5·4-s + (0.0777 − 0.996i)5-s + (0.293 − 0.643i)6-s + (0.871 + 0.491i)7-s + 0.353i·8-s + (0.654 + 0.755i)9-s + (−0.704 − 0.0549i)10-s + (0.519 + 0.899i)11-s + (−0.454 − 0.207i)12-s + (−0.179 + 0.103i)13-s + (0.347 − 0.615i)14-s + (0.484 − 0.874i)15-s + 0.250·16-s + (1.05 + 0.606i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.802 + 0.596i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.802 + 0.596i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.802 + 0.596i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.06227 - 0.682185i\)
\(L(\frac12)\) \(\approx\) \(2.06227 - 0.682185i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.57 - 0.719i)T \)
5 \( 1 + (-0.173 + 2.22i)T \)
7 \( 1 + (-2.30 - 1.29i)T \)
good11 \( 1 + (-1.72 - 2.98i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (0.648 - 0.374i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-4.33 - 2.50i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.11 + 5.38i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.50 - 0.869i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.70 + 8.15i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 7.89T + 31T^{2} \)
37 \( 1 + (-2.60 + 1.50i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.93 + 5.08i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.64 + 3.83i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 - 5.05iT - 47T^{2} \)
53 \( 1 + (-6.61 - 3.81i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 7.38T + 59T^{2} \)
61 \( 1 - 10.7T + 61T^{2} \)
67 \( 1 - 9.47iT - 67T^{2} \)
71 \( 1 - 2.38T + 71T^{2} \)
73 \( 1 + (5.38 + 3.10i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 14.2T + 79T^{2} \)
83 \( 1 + (-3.90 - 2.25i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (7.55 + 13.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.3 + 6.56i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.33012268263828166222530463099, −9.559992183446608378942960218764, −8.862068191102241352730661384307, −8.273143316288378024966833110190, −7.30130348631856012531551441468, −5.54148716286722674335023616923, −4.63378898969658406646696154473, −4.00170368944084028907201645321, −2.44838360038441074731659132778, −1.52975498501985290015264603654, 1.46718540642358827668197453660, 3.10630435396515587371807334368, 3.90810515580026635222285074674, 5.34924960696022505403893869908, 6.51198969122887656795646988582, 7.18121708820052362674257518453, 8.018857243471890640519545667786, 8.588148563326789809356212505959, 9.745059628637184262048437541570, 10.50441141524955895080988816634

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