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Properties

Label 2-630-35.9-c1-0-14
Degree $2$
Conductor $630$
Sign $-0.0667 + 0.997i$
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  + (−0.866 − 0.5i)2-s + (0.499 + 0.866i)4-s + (1.86 − 1.23i)5-s + (−1.73 + 2i)7-s − 0.999i·8-s + (−2.23 + 0.133i)10-s + (−2.5 − 4.33i)11-s − i·13-s + (2.5 − 0.866i)14-s + (−0.5 + 0.866i)16-s + (1.73 − i)17-s + (3.5 − 6.06i)19-s + (1.99 + i)20-s + 5i·22-s + (−2.59 − 1.5i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (0.834 − 0.550i)5-s + (−0.654 + 0.755i)7-s − 0.353i·8-s + (−0.705 + 0.0423i)10-s + (−0.753 − 1.30i)11-s − 0.277i·13-s + (0.668 − 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.420 − 0.242i)17-s + (0.802 − 1.39i)19-s + (0.447 + 0.223i)20-s + 1.06i·22-s + (−0.541 − 0.312i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0667 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.0667 + 0.997i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.0667 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.701263 - 0.749744i\)
\(L(\frac12)\) \(\approx\) \(0.701263 - 0.749744i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 \)
5 \( 1 + (-1.86 + 1.23i)T \)
7 \( 1 + (1.73 - 2i)T \)
good11 \( 1 + (2.5 + 4.33i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + iT - 13T^{2} \)
17 \( 1 + (-1.73 + i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.5 + 6.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (2.59 + 1.5i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.33 - 2.5i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 - 9T + 41T^{2} \)
43 \( 1 + 10iT - 43T^{2} \)
47 \( 1 + (11.2 + 6.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.866 - 0.5i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-5.19 + 3i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 2T + 71T^{2} \)
73 \( 1 + (3.46 - 2i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (7 - 12.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10iT - 83T^{2} \)
89 \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 8iT - 97T^{2} \)
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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

−10.19160946893495040506144692106, −9.502675774549331498379141986042, −8.753470721463080884981135232691, −8.095746143156062731131868243220, −6.75595947008619736698561866764, −5.80261592840217238055039432541, −5.04574005307900156661692512194, −3.22861360073424968778191563834, −2.44304109456564433177194848731, −0.69925457155572616624206529857, 1.58824843346642824685152550092, 2.90026453874013755066336085531, 4.34186435722966545100019252352, 5.71020202249618517939430954686, 6.38891786870038457315112871476, 7.45870583635937468384835465786, 7.86979499082999213797474107311, 9.612424133156071647711089862181, 9.735746345561886869117763040677, 10.43103719965840888221508702155

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