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Properties

Label 2-630-63.38-c1-0-14
Degree $2$
Conductor $630$
Sign $0.986 - 0.165i$
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 + (1.69 + 0.363i)3-s − 4-s + (−0.5 + 0.866i)5-s + (0.363 − 1.69i)6-s + (−1.05 + 2.42i)7-s + i·8-s + (2.73 + 1.23i)9-s + (0.866 + 0.5i)10-s + (4.90 − 2.83i)11-s + (−1.69 − 0.363i)12-s + (−2.43 + 1.40i)13-s + (2.42 + 1.05i)14-s + (−1.16 + 1.28i)15-s + 16-s + (−3.51 + 6.08i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.977 + 0.210i)3-s − 0.5·4-s + (−0.223 + 0.387i)5-s + (0.148 − 0.691i)6-s + (−0.398 + 0.917i)7-s + 0.353i·8-s + (0.911 + 0.410i)9-s + (0.273 + 0.158i)10-s + (1.47 − 0.853i)11-s + (−0.488 − 0.105i)12-s + (−0.675 + 0.389i)13-s + (0.648 + 0.281i)14-s + (−0.299 + 0.331i)15-s + 0.250·16-s + (−0.851 + 1.47i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 - 0.165i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.986 - 0.165i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.986 - 0.165i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.89650 + 0.158080i\)
\(L(\frac12)\) \(\approx\) \(1.89650 + 0.158080i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 0.363i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
7 \( 1 + (1.05 - 2.42i)T \)
good11 \( 1 + (-4.90 + 2.83i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 + (2.43 - 1.40i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (3.51 - 6.08i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.76 + 2.17i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.52 - 2.03i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-5.28 - 3.05i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.368iT - 31T^{2} \)
37 \( 1 + (-0.783 - 1.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (1.39 + 2.41i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.18 + 5.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 5.18T + 47T^{2} \)
53 \( 1 + (-3.02 - 1.74i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + 10.6T + 59T^{2} \)
61 \( 1 + 14.6iT - 61T^{2} \)
67 \( 1 - 2.73T + 67T^{2} \)
71 \( 1 - 9.06iT - 71T^{2} \)
73 \( 1 + (14.3 + 8.29i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 8.74T + 79T^{2} \)
83 \( 1 + (-8.34 + 14.4i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.332 - 0.575i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.68 + 0.971i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.60094391936492868031401987847, −9.581759287337310780825565316991, −8.958944375451303576612972215227, −8.455777992056839778406119974497, −7.11344520898317789933268164739, −6.18804078190150784557905432932, −4.73609564933598057643116715274, −3.64500628413471099288120576099, −2.94070710119233729178062192043, −1.70100705013682502495355005638, 1.07466671662386397854545568718, 2.91616852356125294340568813714, 4.16316668696665427866103269189, 4.75853558789357852541360630437, 6.49372939929439079621641184492, 7.16048413521819973422228124303, 7.71282529775113682329967246492, 8.877776583485558729145067875559, 9.507290073493219608911613403490, 10.08250994852688989208261119066

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