Properties

Label 2-630-105.23-c1-0-1
Degree $2$
Conductor $630$
Sign $-0.761 - 0.647i$
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  + (−0.965 + 0.258i)2-s + (0.866 − 0.499i)4-s + (0.915 + 2.04i)5-s + (−2.36 − 1.18i)7-s + (−0.707 + 0.707i)8-s + (−1.41 − 1.73i)10-s + (−1.10 + 0.636i)11-s + (2.15 + 2.15i)13-s + (2.59 + 0.537i)14-s + (0.500 − 0.866i)16-s + (−1.55 + 5.80i)17-s + (−6.20 − 3.58i)19-s + (1.81 + 1.30i)20-s + (0.900 − 0.900i)22-s + (1.11 + 4.14i)23-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.433 − 0.249i)4-s + (0.409 + 0.912i)5-s + (−0.893 − 0.449i)7-s + (−0.249 + 0.249i)8-s + (−0.446 − 0.548i)10-s + (−0.332 + 0.191i)11-s + (0.596 + 0.596i)13-s + (0.692 + 0.143i)14-s + (0.125 − 0.216i)16-s + (−0.376 + 1.40i)17-s + (−1.42 − 0.822i)19-s + (0.405 + 0.292i)20-s + (0.191 − 0.191i)22-s + (0.231 + 0.864i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.761 - 0.647i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.761 - 0.647i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (233, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.761 - 0.647i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.213638 + 0.581035i\)
\(L(\frac12)\) \(\approx\) \(0.213638 + 0.581035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 \)
5 \( 1 + (-0.915 - 2.04i)T \)
7 \( 1 + (2.36 + 1.18i)T \)
good11 \( 1 + (1.10 - 0.636i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (-2.15 - 2.15i)T + 13iT^{2} \)
17 \( 1 + (1.55 - 5.80i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (6.20 + 3.58i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.11 - 4.14i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 - 1.25T + 29T^{2} \)
31 \( 1 + (2.35 + 4.08i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.535 - 1.99i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.655iT - 41T^{2} \)
43 \( 1 + (-7.20 - 7.20i)T + 43iT^{2} \)
47 \( 1 + (10.4 - 2.80i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (13.2 + 3.55i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.688 - 1.19i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (2.57 - 4.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (4.19 + 1.12i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.159iT - 71T^{2} \)
73 \( 1 + (2.65 - 9.90i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-9.23 - 5.33i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (6.09 - 6.09i)T - 83iT^{2} \)
89 \( 1 + (-3.79 + 6.57i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-10.9 + 10.9i)T - 97iT^{2} \)
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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

−10.90895835745322691545577652292, −9.991949501924765288807781681732, −9.352643516259875345664457421466, −8.347795411966844612704044978831, −7.31657486999646022758514794839, −6.44460446342485071384937042896, −6.06009627898597712767638473309, −4.28812529188522612025254298501, −3.09903515053502189740018028804, −1.84140122348104058004812161784, 0.39786501185102052878022706200, 2.10096432162379085338302747663, 3.28834572747673824573976884250, 4.74033350334917663666004076401, 5.86949079417863728977741488382, 6.61015790851782025508531766526, 7.896729350200490991624999744569, 8.760345197018250898653093077284, 9.227657931291911767002515703101, 10.20099102367061387547195479405

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