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Properties

Label 2-630-315.214-c1-0-44
Degree $2$
Conductor $630$
Sign $-0.984 + 0.176i$
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 + (1.32 + 1.11i)3-s − 4-s + (−1.69 − 1.45i)5-s + (1.11 − 1.32i)6-s + (−2.63 − 0.270i)7-s + i·8-s + (0.533 + 2.95i)9-s + (−1.45 + 1.69i)10-s + (−1.73 − 3.00i)11-s + (−1.32 − 1.11i)12-s + (1.09 − 0.633i)13-s + (−0.270 + 2.63i)14-s + (−0.643 − 3.81i)15-s + 16-s + (−4.66 − 2.69i)17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.767 + 0.641i)3-s − 0.5·4-s + (−0.759 − 0.650i)5-s + (0.453 − 0.542i)6-s + (−0.994 − 0.102i)7-s + 0.353i·8-s + (0.177 + 0.984i)9-s + (−0.459 + 0.537i)10-s + (−0.523 − 0.906i)11-s + (−0.383 − 0.320i)12-s + (0.304 − 0.175i)13-s + (−0.0723 + 0.703i)14-s + (−0.166 − 0.986i)15-s + 0.250·16-s + (−1.13 − 0.652i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $-0.984 + 0.176i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ -0.984 + 0.176i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0525843 - 0.590379i\)
\(L(\frac12)\) \(\approx\) \(0.0525843 - 0.590379i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 1.11i)T \)
5 \( 1 + (1.69 + 1.45i)T \)
7 \( 1 + (2.63 + 0.270i)T \)
good11 \( 1 + (1.73 + 3.00i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.09 + 0.633i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + (4.66 + 2.69i)T + (8.5 + 14.7i)T^{2} \)
19 \( 1 + (2.23 + 3.87i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.15 + 2.39i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.04 + 1.81i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 0.803T + 31T^{2} \)
37 \( 1 + (5.27 - 3.04i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.28 - 3.96i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.972 - 0.561i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + 9.74iT - 47T^{2} \)
53 \( 1 + (6.88 + 3.97i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 - 12.1T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 - 1.64iT - 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 + (-3.28 - 1.89i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + 1.59T + 79T^{2} \)
83 \( 1 + (-9.12 - 5.26i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (4.84 + 8.38i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7.48 - 4.32i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.20664306913338637155946791779, −9.252742690444737856290731982471, −8.657383172730067965249024555834, −7.977720738523596302868958440297, −6.65103448305298134305810946987, −5.19568979710285239288323189290, −4.26958427429598390257328583936, −3.42627205410893587263577732118, −2.48303173515625833166802795025, −0.27792161955487451683327185002, 2.18621881087650390969705639314, 3.48752415716414053777232765313, 4.26750369736456401384644103091, 6.04348327170021229498747583177, 6.68606466909837900721762173707, 7.45865886869793969290574500553, 8.174169892350842067936341046427, 9.051590979605015475728963567673, 9.960730638130931797375231217402, 10.84828879888922801672605868257

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