↑

Properties

Label 2-630-315.194-c1-0-0
Degree $2$
Conductor $630$
Sign $0.511 - 0.859i$
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  − 2-s + (−1.34 − 1.09i)3-s + 4-s + (−1.79 − 1.33i)5-s + (1.34 + 1.09i)6-s + (−2.61 − 0.389i)7-s − 8-s + (0.600 + 2.93i)9-s + (1.79 + 1.33i)10-s + (−1.98 − 1.14i)11-s + (−1.34 − 1.09i)12-s + (2.02 − 3.49i)13-s + (2.61 + 0.389i)14-s + (0.950 + 3.75i)15-s + 16-s + (−5.58 + 3.22i)17-s + ⋯
L(s)  = 1  − 0.707·2-s + (−0.774 − 0.632i)3-s + 0.5·4-s + (−0.803 − 0.595i)5-s + (0.547 + 0.447i)6-s + (−0.989 − 0.147i)7-s − 0.353·8-s + (0.200 + 0.979i)9-s + (0.567 + 0.421i)10-s + (−0.597 − 0.344i)11-s + (−0.387 − 0.316i)12-s + (0.560 − 0.970i)13-s + (0.699 + 0.104i)14-s + (0.245 + 0.969i)15-s + 0.250·16-s + (−1.35 + 0.782i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.511 - 0.859i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.198305 + 0.112744i\)
\(L(\frac12)\) \(\approx\) \(0.198305 + 0.112744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + (1.34 + 1.09i)T \)
5 \( 1 + (1.79 + 1.33i)T \)
7 \( 1 + (2.61 + 0.389i)T \)
good11 \( 1 + (1.98 + 1.14i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.02 + 3.49i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (5.58 - 3.22i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.177 + 0.102i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.00 - 1.73i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-2.95 + 1.70i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.975iT - 31T^{2} \)
37 \( 1 + (-7.40 - 4.27i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.15 - 2.00i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.12 + 1.22i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.77iT - 47T^{2} \)
53 \( 1 + (-2.35 - 4.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 9.07T + 59T^{2} \)
61 \( 1 - 10.7iT - 61T^{2} \)
67 \( 1 - 6.05iT - 67T^{2} \)
71 \( 1 - 11.0iT - 71T^{2} \)
73 \( 1 + (-7.78 - 13.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + 7.23T + 79T^{2} \)
83 \( 1 + (-9.91 + 5.72i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5.08 + 8.81i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.24 + 7.34i)T + (-48.5 + 84.0i)T^{2} \)
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

−10.79048487126749004530468894269, −10.03724648844957114944211116200, −8.762119308510588917465708285499, −8.149443837199538033654358284467, −7.27243495729406563325523343682, −6.36713651919806671175487427514, −5.54713034284317312892671636242, −4.20182487593830696000672393293, −2.78952749327049765243262134359, −0.984803122877983221496786463940, 0.21856234945606530635425238485, 2.61262934940980275151718680313, 3.81236651805755000205801788279, 4.82008724472955078113428286226, 6.39160841580140371387886255641, 6.65888519431512110724528747448, 7.76604237609479391826626609598, 9.070001061043938547995252232907, 9.495895508799572335698356328147, 10.68648069447810884899658079431

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