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Properties

Label 2-630-315.194-c1-0-39
Degree $2$
Conductor $630$
Sign $0.908 + 0.416i$
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 + (1.70 − 0.317i)3-s + 4-s + (1.22 − 1.86i)5-s + (1.70 − 0.317i)6-s + (−0.691 + 2.55i)7-s + 8-s + (2.79 − 1.08i)9-s + (1.22 − 1.86i)10-s + (−3.10 − 1.79i)11-s + (1.70 − 0.317i)12-s + (2.03 − 3.51i)13-s + (−0.691 + 2.55i)14-s + (1.49 − 3.57i)15-s + 16-s + (−4.57 + 2.63i)17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.983 − 0.183i)3-s + 0.5·4-s + (0.549 − 0.835i)5-s + (0.695 − 0.129i)6-s + (−0.261 + 0.965i)7-s + 0.353·8-s + (0.932 − 0.360i)9-s + (0.388 − 0.590i)10-s + (−0.936 − 0.540i)11-s + (0.491 − 0.0917i)12-s + (0.563 − 0.975i)13-s + (−0.184 + 0.682i)14-s + (0.387 − 0.921i)15-s + 0.250·16-s + (−1.10 + 0.640i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.908 + 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.908 + 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.908 + 0.416i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (509, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.908 + 0.416i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.14462 - 0.686710i\)
\(L(\frac12)\) \(\approx\) \(3.14462 - 0.686710i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.70 + 0.317i)T \)
5 \( 1 + (-1.22 + 1.86i)T \)
7 \( 1 + (0.691 - 2.55i)T \)
good11 \( 1 + (3.10 + 1.79i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-2.03 + 3.51i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.57 - 2.63i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.13 - 1.81i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.39 - 4.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.99 - 4.03i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 3.52iT - 31T^{2} \)
37 \( 1 + (0.879 + 0.507i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.80 - 8.31i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.43 + 2.56i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 11.5iT - 47T^{2} \)
53 \( 1 + (5.71 + 9.89i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 - 7.03T + 59T^{2} \)
61 \( 1 - 11.2iT - 61T^{2} \)
67 \( 1 - 6.99iT - 67T^{2} \)
71 \( 1 - 0.828iT - 71T^{2} \)
73 \( 1 + (-3.51 - 6.08i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 0.0299T + 79T^{2} \)
83 \( 1 + (6.89 - 3.98i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.625 - 1.08i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.69 + 8.13i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.46059400874411077187942001157, −9.552588974460709709787683497388, −8.613394569886656811881971413833, −8.181187485661335564822906141034, −6.91262154831405913878997659044, −5.68693742934174825271185249572, −5.22738321743863918151475569658, −3.71304272195583670592723878618, −2.79846180993241808319060635762, −1.65111976081761429538555984503, 2.02528819823801437803608986053, 2.91835388625237575834685491721, 4.00126391142719037326124918937, 4.84767675820876780328352331550, 6.32663079845700859447179396461, 7.13688483985315040327805027281, 7.67627269527129554401291045314, 9.156491120016535915887013822719, 9.764016317572805048174440116438, 10.77988492621553400176819512803

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