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Properties

Label 2-630-315.209-c1-0-12
Degree $2$
Conductor $630$
Sign $-0.937 - 0.348i$
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  + (−0.5 + 0.866i)2-s + (−0.979 + 1.42i)3-s + (−0.499 − 0.866i)4-s + (−0.452 + 2.18i)5-s + (−0.747 − 1.56i)6-s + (2.48 + 0.896i)7-s + 0.999·8-s + (−1.08 − 2.79i)9-s + (−1.67 − 1.48i)10-s + (5.48 + 3.16i)11-s + (1.72 + 0.134i)12-s + (2.28 + 3.95i)13-s + (−2.02 + 1.70i)14-s + (−2.68 − 2.79i)15-s + (−0.5 + 0.866i)16-s + 2.72i·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.565 + 0.824i)3-s + (−0.249 − 0.433i)4-s + (−0.202 + 0.979i)5-s + (−0.304 − 0.637i)6-s + (0.940 + 0.339i)7-s + 0.353·8-s + (−0.360 − 0.932i)9-s + (−0.528 − 0.470i)10-s + (1.65 + 0.953i)11-s + (0.498 + 0.0387i)12-s + (0.632 + 1.09i)13-s + (−0.540 + 0.456i)14-s + (−0.693 − 0.720i)15-s + (−0.125 + 0.216i)16-s + 0.661i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.937 - 0.348i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.937 - 0.348i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.937 - 0.348i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.197972 + 1.10090i\)
\(L(\frac12)\) \(\approx\) \(0.197972 + 1.10090i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 0.866i)T \)
3 \( 1 + (0.979 - 1.42i)T \)
5 \( 1 + (0.452 - 2.18i)T \)
7 \( 1 + (-2.48 - 0.896i)T \)
good11 \( 1 + (-5.48 - 3.16i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.28 - 3.95i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 2.72iT - 17T^{2} \)
19 \( 1 + 4.11iT - 19T^{2} \)
23 \( 1 + (-1.15 - 2.00i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.173 + 0.100i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.10 - 1.21i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 10.4iT - 37T^{2} \)
41 \( 1 + (-1.03 - 1.78i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (7.26 + 4.19i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (7.86 + 4.53i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 6.77T + 53T^{2} \)
59 \( 1 + (-0.616 - 1.06i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.66 + 0.961i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.07 - 4.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 8.31iT - 71T^{2} \)
73 \( 1 - 13.9T + 73T^{2} \)
79 \( 1 + (-1.01 + 1.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (3.64 + 2.10i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + 9.48T + 89T^{2} \)
97 \( 1 + (2.42 - 4.20i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.13449328975732292245891693745, −10.02383611310783005793361174796, −9.208106273580177765795331455005, −8.588617521759724329675622350975, −7.12735308592980842100375021672, −6.66024505863287926578324558800, −5.66144679202573997064671319178, −4.46597534724312307098525662357, −3.79143424588829713325737992065, −1.76382235097924151683556336410, 0.869822980895720114102396421008, 1.53075225101590475081545381042, 3.42713892614835979709939244703, 4.61798326155645906004168657406, 5.60501503532254195578649258273, 6.64007546519750600056548121475, 8.011764830002430099585537784390, 8.247450652439178557070844690601, 9.230010948573039532036593052411, 10.44090059780628325392631858510

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