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Properties

Label 2-630-63.5-c1-0-30
Degree $2$
Conductor $630$
Sign $-0.906 - 0.422i$
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 + (0.661 − 1.60i)3-s − 4-s + (0.5 + 0.866i)5-s + (−1.60 − 0.661i)6-s + (−2.57 − 0.589i)7-s + i·8-s + (−2.12 − 2.11i)9-s + (0.866 − 0.5i)10-s + (−2.51 − 1.45i)11-s + (−0.661 + 1.60i)12-s + (1.69 + 0.979i)13-s + (−0.589 + 2.57i)14-s + (1.71 − 0.227i)15-s + 16-s + (−1.68 − 2.91i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.381 − 0.924i)3-s − 0.5·4-s + (0.223 + 0.387i)5-s + (−0.653 − 0.269i)6-s + (−0.974 − 0.222i)7-s + 0.353i·8-s + (−0.708 − 0.705i)9-s + (0.273 − 0.158i)10-s + (−0.759 − 0.438i)11-s + (−0.190 + 0.462i)12-s + (0.470 + 0.271i)13-s + (−0.157 + 0.689i)14-s + (0.443 − 0.0588i)15-s + 0.250·16-s + (−0.408 − 0.707i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 - 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.906 - 0.422i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.906 - 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.171246 + 0.771918i\)
\(L(\frac12)\) \(\approx\) \(0.171246 + 0.771918i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.661 + 1.60i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.57 + 0.589i)T \)
good11 \( 1 + (2.51 + 1.45i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.69 - 0.979i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.68 + 2.91i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (6.97 + 4.02i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.47 - 1.42i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.87 + 1.08i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 2.20iT - 31T^{2} \)
37 \( 1 + (-1.68 + 2.91i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-4.24 + 7.35i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.991 - 1.71i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 0.647T + 47T^{2} \)
53 \( 1 + (-9.25 + 5.34i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 - 9.11T + 59T^{2} \)
61 \( 1 - 3.03iT - 61T^{2} \)
67 \( 1 + 15.1T + 67T^{2} \)
71 \( 1 + 7.94iT - 71T^{2} \)
73 \( 1 + (12.5 - 7.26i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 4.29T + 79T^{2} \)
83 \( 1 + (-6.87 - 11.9i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-8.43 + 14.6i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.21 + 3.01i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.23657964993877297210446701993, −9.141421778653372391610768574093, −8.579722159781001055312969224495, −7.38052290614591344673310187306, −6.57725502146159658818474601551, −5.71753739872879808782760998524, −4.11320179075536684085977094062, −2.96424351706365928254033642592, −2.21710626433021241237621863665, −0.38416269777150920941556984950, 2.42056142859948357047653628715, 3.78867767779490165377672047829, 4.59324549696705135390358924274, 5.78422743228377264587236299544, 6.35058107134192725008080744900, 7.82577902283379484844072089894, 8.532113131265279739303768427890, 9.221361048966609016065180563256, 10.26619408765428073133539611608, 10.49801233735445763663298126179

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