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Properties

Label 2-630-63.38-c1-0-29
Degree $2$
Conductor $630$
Sign $-0.591 + 0.805i$
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 + (−0.0386 − 1.73i)3-s − 4-s + (−0.5 + 0.866i)5-s + (1.73 − 0.0386i)6-s + (0.281 − 2.63i)7-s − i·8-s + (−2.99 + 0.133i)9-s + (−0.866 − 0.5i)10-s + (0.390 − 0.225i)11-s + (0.0386 + 1.73i)12-s + (−4.26 + 2.46i)13-s + (2.63 + 0.281i)14-s + (1.51 + 0.832i)15-s + 16-s + (3.93 − 6.81i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.0223 − 0.999i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (0.706 − 0.0157i)6-s + (0.106 − 0.994i)7-s − 0.353i·8-s + (−0.999 + 0.0446i)9-s + (−0.273 − 0.158i)10-s + (0.117 − 0.0679i)11-s + (0.0111 + 0.499i)12-s + (−1.18 + 0.682i)13-s + (0.703 + 0.0753i)14-s + (0.392 + 0.214i)15-s + 0.250·16-s + (0.953 − 1.65i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.591 + 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.591 + 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.591 + 0.805i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.591 + 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.275268 - 0.543690i\)
\(L(\frac12)\) \(\approx\) \(0.275268 - 0.543690i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.0386 + 1.73i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (-0.281 + 2.63i)T \)
good11 \( 1 + (-0.390 + 0.225i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (4.26 - 2.46i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (-3.93 + 6.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (4.75 - 2.74i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.21 + 2.43i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (8.05 + 4.65i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.574iT - 31T^{2} \)
37 \( 1 + (0.721 + 1.25i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (0.956 + 1.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.459 - 0.795i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 7.42T + 47T^{2} \)
53 \( 1 + (8.30 + 4.79i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 8.86T + 59T^{2} \)
61 \( 1 + 9.86iT - 61T^{2} \)
67 \( 1 + 4.64T + 67T^{2} \)
71 \( 1 - 3.88iT - 71T^{2} \)
73 \( 1 + (-4.91 - 2.83i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 2.00T + 79T^{2} \)
83 \( 1 + (-4.76 + 8.25i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-1.98 - 3.43i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-8.69 - 5.01i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.16971688735765743574960498090, −9.355816109321080957048863800808, −8.105709858341998611229572074923, −7.47024162863283043269233318376, −6.95968627664699384438606918937, −6.02902362600965073560233640906, −4.86087112487420530975645429485, −3.69130154395825385721833875019, −2.17061980457980475577486263790, −0.31652045129475371705901788403, 2.06760396654355853862541942627, 3.29644929585258710301009009200, 4.27781263967789859290951975743, 5.28265476685957625738967329968, 5.93655667459177945579204482238, 7.76559323367222446096811084650, 8.577468496038160929760852410406, 9.283469873066321793507517855107, 10.10779041393271614077470186845, 10.77539895869565238879110340034

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