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Properties

Label 2-630-315.184-c1-0-45
Degree $2$
Conductor $630$
Sign $-0.998 + 0.0467i$
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 + (0.286 − 1.70i)3-s − 4-s + (0.867 − 2.06i)5-s + (−1.70 − 0.286i)6-s + (−0.0441 − 2.64i)7-s + i·8-s + (−2.83 − 0.979i)9-s + (−2.06 − 0.867i)10-s + (2.13 − 3.69i)11-s + (−0.286 + 1.70i)12-s + (5.34 + 3.08i)13-s + (−2.64 + 0.0441i)14-s + (−3.27 − 2.07i)15-s + 16-s + (0.933 − 0.539i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.165 − 0.986i)3-s − 0.5·4-s + (0.388 − 0.921i)5-s + (−0.697 − 0.117i)6-s + (−0.0166 − 0.999i)7-s + 0.353i·8-s + (−0.945 − 0.326i)9-s + (−0.651 − 0.274i)10-s + (0.643 − 1.11i)11-s + (−0.0828 + 0.493i)12-s + (1.48 + 0.855i)13-s + (−0.707 + 0.0117i)14-s + (−0.844 − 0.535i)15-s + 0.250·16-s + (0.226 − 0.130i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0467i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.998 + 0.0467i$
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: \((2,\ 630,\ (\ :1/2),\ -0.998 + 0.0467i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0369065 - 1.57699i\)
\(L(\frac12)\) \(\approx\) \(0.0369065 - 1.57699i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.286 + 1.70i)T \)
5 \( 1 + (-0.867 + 2.06i)T \)
7 \( 1 + (0.0441 + 2.64i)T \)
good11 \( 1 + (-2.13 + 3.69i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.34 - 3.08i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-0.933 + 0.539i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.68 - 2.92i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3.32 - 1.92i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.28 - 3.96i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 4.56T + 31T^{2} \)
37 \( 1 + (-10.2 - 5.90i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-3.08 + 5.34i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.94 - 1.12i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.04iT - 47T^{2} \)
53 \( 1 + (-6.03 + 3.48i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 1.38T + 59T^{2} \)
61 \( 1 + 8.69T + 61T^{2} \)
67 \( 1 - 2.47iT - 67T^{2} \)
71 \( 1 - 4.28T + 71T^{2} \)
73 \( 1 + (10.8 - 6.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + 14.4T + 79T^{2} \)
83 \( 1 + (-13.9 + 8.06i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (1.06 - 1.83i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-11.5 + 6.65i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.24575358098762941527257524671, −9.094870318479269827498756704286, −8.600192103383537942521335624795, −7.73607204569108244396600073961, −6.40658366617477494208600551235, −5.79444027792285060698936060774, −4.24258133371167789901521329919, −3.41472459410995581781990352567, −1.66722512810460524665290650559, −0.942544114049128285053741156777, 2.36298948417954666847612750791, 3.56176991007511039760043101221, 4.58744236483411769667306908745, 5.89386467930019768707366627784, 6.17795320012106353056348213941, 7.54852314119426948983887118497, 8.527851273021945652109572105288, 9.282048229363028284634958921812, 9.994666897517597068313279542968, 10.83217625342603083018747187492

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