Properties

Label 2-630-315.184-c1-0-44
Degree $2$
Conductor $630$
Sign $-0.131 - 0.991i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  i·2-s + (−1.72 + 0.168i)3-s − 4-s + (−1.95 − 1.08i)5-s + (0.168 + 1.72i)6-s + (0.831 − 2.51i)7-s + i·8-s + (2.94 − 0.580i)9-s + (−1.08 + 1.95i)10-s + (0.171 − 0.296i)11-s + (1.72 − 0.168i)12-s + (−4.33 − 2.50i)13-s + (−2.51 − 0.831i)14-s + (3.55 + 1.53i)15-s + 16-s + (−0.981 + 0.566i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (−0.995 + 0.0972i)3-s − 0.5·4-s + (−0.875 − 0.483i)5-s + (0.0687 + 0.703i)6-s + (0.314 − 0.949i)7-s + 0.353i·8-s + (0.981 − 0.193i)9-s + (−0.341 + 0.618i)10-s + (0.0516 − 0.0895i)11-s + (0.497 − 0.0486i)12-s + (−1.20 − 0.694i)13-s + (−0.671 − 0.222i)14-s + (0.918 + 0.396i)15-s + 0.250·16-s + (−0.238 + 0.137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.131 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.131 - 0.991i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.131 - 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0111896 + 0.0127718i\)
\(L(\frac12)\) \(\approx\) \(0.0111896 + 0.0127718i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.72 - 0.168i)T \)
5 \( 1 + (1.95 + 1.08i)T \)
7 \( 1 + (-0.831 + 2.51i)T \)
good11 \( 1 + (-0.171 + 0.296i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.33 + 2.50i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (0.981 - 0.566i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0114 - 0.0197i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6.65 - 3.83i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.68 - 4.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.19T + 31T^{2} \)
37 \( 1 + (-6.66 - 3.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.52 - 4.36i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.27 - 1.89i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 5.35iT - 47T^{2} \)
53 \( 1 + (3.77 - 2.18i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 6.85T + 59T^{2} \)
61 \( 1 + 12.4T + 61T^{2} \)
67 \( 1 + 10.4iT - 67T^{2} \)
71 \( 1 + 7.78T + 71T^{2} \)
73 \( 1 + (-11.7 + 6.80i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (-0.917 + 0.529i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-3.36 + 5.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-15.3 + 8.86i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.19993390950671798217500131213, −9.438706692991954322706965927757, −7.997561675743525686712480406811, −7.53243303654617689822938763444, −6.25141491382240811290259209664, −4.88945721707226762875356143593, −4.49634733007173111892003706192, −3.33195144364358900217895203674, −1.34025532817720711666892430389, −0.01133462689507369654337990317, 2.34428622437080960112148804873, 4.18958939269574228177525427831, 4.84607487448758422336482116921, 5.98186741513860196934398123558, 6.71203778990275692046957406508, 7.57771448198055217921631974632, 8.350794107756931201457553841754, 9.529741808439869025991552358495, 10.35407250067202130711256686141, 11.43494762122561897956431476082

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