↑

Properties

Label 2-630-35.13-c1-0-13
Degree $2$
Conductor $630$
Sign $-0.0347 + 0.999i$
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  + (0.707 − 0.707i)2-s − 1.00i·4-s + (−1.52 + 1.63i)5-s + (−1.23 − 2.33i)7-s + (−0.707 − 0.707i)8-s + (0.0743 + 2.23i)10-s + 5.73·11-s + (3.41 − 3.41i)13-s + (−2.52 − 0.781i)14-s − 1.00·16-s + (−2.57 − 2.57i)17-s + 1.85·19-s + (1.63 + 1.52i)20-s + (4.05 − 4.05i)22-s + (−6.46 − 6.46i)23-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.683 + 0.730i)5-s + (−0.466 − 0.884i)7-s + (−0.250 − 0.250i)8-s + (0.0234 + 0.706i)10-s + 1.72·11-s + (0.946 − 0.946i)13-s + (−0.675 − 0.208i)14-s − 0.250·16-s + (−0.624 − 0.624i)17-s + 0.424·19-s + (0.365 + 0.341i)20-s + (0.864 − 0.864i)22-s + (−1.34 − 1.34i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0347 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0347 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0347 + 0.999i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (433, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0347 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.14803 - 1.18864i\)
\(L(\frac12)\) \(\approx\) \(1.14803 - 1.18864i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 \)
5 \( 1 + (1.52 - 1.63i)T \)
7 \( 1 + (1.23 + 2.33i)T \)
good11 \( 1 - 5.73T + 11T^{2} \)
13 \( 1 + (-3.41 + 3.41i)T - 13iT^{2} \)
17 \( 1 + (2.57 + 2.57i)T + 17iT^{2} \)
19 \( 1 - 1.85T + 19T^{2} \)
23 \( 1 + (6.46 + 6.46i)T + 23iT^{2} \)
29 \( 1 + 3.47iT - 29T^{2} \)
31 \( 1 + 0.469iT - 31T^{2} \)
37 \( 1 + (-0.574 + 0.574i)T - 37iT^{2} \)
41 \( 1 + 1.03iT - 41T^{2} \)
43 \( 1 + (-6.17 - 6.17i)T + 43iT^{2} \)
47 \( 1 + (-3.85 - 3.85i)T + 47iT^{2} \)
53 \( 1 + (-1.85 - 1.85i)T + 53iT^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 8.53iT - 61T^{2} \)
67 \( 1 + (4.46 - 4.46i)T - 67iT^{2} \)
71 \( 1 + 5.13T + 71T^{2} \)
73 \( 1 + (-7.80 + 7.80i)T - 73iT^{2} \)
79 \( 1 - 4.16iT - 79T^{2} \)
83 \( 1 + (1.77 - 1.77i)T - 83iT^{2} \)
89 \( 1 + 9.12T + 89T^{2} \)
97 \( 1 + (0.0119 + 0.0119i)T + 97iT^{2} \)
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

−10.53742873579485408972057328318, −9.785447183377840324655206935322, −8.699754814727703626873186308188, −7.58512363550126800448295745594, −6.62504547886819012651689510628, −6.04083313617350312969240907022, −4.22770498119929956341995241052, −3.88174229477348259726616740533, −2.73061556599519458384176666158, −0.840559680775090187968352606714, 1.69419999305939079951124337237, 3.68515771865377441957743017960, 4.07249899712044638069370938166, 5.45738186284096887059011957673, 6.28051585974583154912154601194, 7.08296819801289243922816960667, 8.351369093507204863550647069680, 8.931359899236795706820022480661, 9.553376370413129797369192447441, 11.24397322041015359823808375947

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