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Properties

Label 2-630-63.58-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.781 - 0.623i$
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  + 2-s + (−1.5 + 0.866i)3-s + 4-s + (0.5 + 0.866i)5-s + (−1.5 + 0.866i)6-s + (−2.62 + 0.358i)7-s + 8-s + (1.5 − 2.59i)9-s + (0.5 + 0.866i)10-s + (−2.12 + 3.67i)11-s + (−1.5 + 0.866i)12-s + (−1 + 1.73i)13-s + (−2.62 + 0.358i)14-s + (−1.5 − 0.866i)15-s + 16-s + ⋯
L(s)  = 1  + 0.707·2-s + (−0.866 + 0.499i)3-s + 0.5·4-s + (0.223 + 0.387i)5-s + (−0.612 + 0.353i)6-s + (−0.990 + 0.135i)7-s + 0.353·8-s + (0.5 − 0.866i)9-s + (0.158 + 0.273i)10-s + (−0.639 + 1.10i)11-s + (−0.433 + 0.249i)12-s + (−0.277 + 0.480i)13-s + (−0.700 + 0.0958i)14-s + (−0.387 − 0.223i)15-s + 0.250·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.781 - 0.623i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.781 - 0.623i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.781 - 0.623i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.332271 + 0.949440i\)
\(L(\frac12)\) \(\approx\) \(0.332271 + 0.949440i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (2.62 - 0.358i)T \)
good11 \( 1 + (2.12 - 3.67i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (1 - 1.73i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.12 - 5.40i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.24 + 7.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.62 - 6.27i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 + (0.121 - 0.210i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 7.24T + 47T^{2} \)
53 \( 1 + (5.12 + 8.87i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 10.2T + 59T^{2} \)
61 \( 1 - 4.48T + 61T^{2} \)
67 \( 1 - 10.4T + 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 + (-6.24 - 10.8i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 14.7T + 79T^{2} \)
83 \( 1 + (-4.5 - 7.79i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.24 + 9.08i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.85841677437026094774482133827, −10.08429464469639334479042386522, −9.752691543725016670876711366614, −8.242091775726980079939021218170, −6.74636387326027972868716480144, −6.55867219498931389760065422526, −5.41222437166460181493742530735, −4.51941725889299800964792239875, −3.52766115861853087528864958122, −2.16911008402207231928102689453, 0.45156741339020191454322610194, 2.32563700120707682699322039336, 3.60784697836687808586014191675, 4.93343564920020681877971557824, 5.74066841585952262322010629659, 6.37403069628029111972849825689, 7.35452537391707474234089044469, 8.307026576669372557759246110324, 9.612741042268169041315876626778, 10.47919671108382050454992476844

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