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Properties

Label 2-630-105.2-c1-0-10
Degree $2$
Conductor $630$
Sign $0.695 - 0.718i$
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  + (0.258 + 0.965i)2-s + (−0.866 + 0.499i)4-s + (1.99 + 1.01i)5-s + (1.78 − 1.94i)7-s + (−0.707 − 0.707i)8-s + (−0.469 + 2.18i)10-s + (2.64 − 1.52i)11-s + (1 − i)13-s + (2.34 + 1.22i)14-s + (0.500 − 0.866i)16-s + (4.16 + 1.11i)17-s + (−5.47 − 3.15i)19-s + (−2.23 + 0.111i)20-s + (2.15 + 2.15i)22-s + (−1.62 + 0.435i)23-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.433 + 0.249i)4-s + (0.889 + 0.456i)5-s + (0.676 − 0.736i)7-s + (−0.249 − 0.249i)8-s + (−0.148 + 0.691i)10-s + (0.797 − 0.460i)11-s + (0.277 − 0.277i)13-s + (0.626 + 0.327i)14-s + (0.125 − 0.216i)16-s + (1.01 + 0.270i)17-s + (−1.25 − 0.724i)19-s + (−0.499 + 0.0250i)20-s + (0.460 + 0.460i)22-s + (−0.339 + 0.0908i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.695 - 0.718i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.695 - 0.718i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (107, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.695 - 0.718i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.84968 + 0.783384i\)
\(L(\frac12)\) \(\approx\) \(1.84968 + 0.783384i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 - 0.965i)T \)
3 \( 1 \)
5 \( 1 + (-1.99 - 1.01i)T \)
7 \( 1 + (-1.78 + 1.94i)T \)
good11 \( 1 + (-2.64 + 1.52i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-1 + i)T - 13iT^{2} \)
17 \( 1 + (-4.16 - 1.11i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (5.47 + 3.15i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (1.62 - 0.435i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 5.88T + 29T^{2} \)
31 \( 1 + (-1.5 - 2.59i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (6.83 - 1.83i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 + (7.63 - 7.63i)T - 43iT^{2} \)
47 \( 1 + (0.0819 + 0.305i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.422 + 1.57i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-5.99 - 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.15 - 5.47i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.89 + 14.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 1.86iT - 71T^{2} \)
73 \( 1 + (-3.66 - 0.982i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.14 + 1.81i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.67 + 8.67i)T + 83iT^{2} \)
89 \( 1 + (6.32 - 10.9i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.84 + 5.84i)T + 97iT^{2} \)
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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

−10.55403843696124694449260716991, −9.923681146539232201238110788418, −8.753835625800769429583658012807, −8.102165692272137329447480057551, −6.90858045373074968851267220605, −6.35701343608276120312676154867, −5.33967528341617955844350245092, −4.30116013430940766285760939030, −3.11853361634094626447195296881, −1.39551465027599792829206915480, 1.46757125217839554250613255194, 2.31539437942382539874679825971, 3.87590152517956898986283148742, 4.92717030848235312821419457504, 5.74614733464198886748678980563, 6.68640907681260490364820888992, 8.276649636633579951500903911281, 8.795764114784914596827518932686, 9.776863218513097010443077595358, 10.35733232098388595872205951831

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