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Properties

Label 2-630-315.184-c1-0-28
Degree $2$
Conductor $630$
Sign $0.863 - 0.503i$
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 + 0.340i)3-s − 4-s + (2.18 + 0.467i)5-s + (−0.340 − 1.69i)6-s + (2.41 + 1.07i)7-s − i·8-s + (2.76 − 1.15i)9-s + (−0.467 + 2.18i)10-s + (2.98 − 5.16i)11-s + (1.69 − 0.340i)12-s + (−5.24 − 3.02i)13-s + (−1.07 + 2.41i)14-s + (−3.87 − 0.0489i)15-s + 16-s + (1.10 − 0.636i)17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.980 + 0.196i)3-s − 0.5·4-s + (0.977 + 0.209i)5-s + (−0.139 − 0.693i)6-s + (0.913 + 0.406i)7-s − 0.353i·8-s + (0.922 − 0.385i)9-s + (−0.147 + 0.691i)10-s + (0.899 − 1.55i)11-s + (0.490 − 0.0983i)12-s + (−1.45 − 0.839i)13-s + (−0.287 + 0.645i)14-s + (−0.999 − 0.0126i)15-s + 0.250·16-s + (0.267 − 0.154i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.503i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(630\)    =    \(2 \cdot 3^{2} \cdot 5 \cdot 7\)
Sign: $0.863 - 0.503i$
Analytic conductor: \(5.03057\)
Root analytic conductor: \(2.24289\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{630} (499, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 630,\ (\ :1/2),\ 0.863 - 0.503i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.31382 + 0.354961i\)
\(L(\frac12)\) \(\approx\) \(1.31382 + 0.354961i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.69 - 0.340i)T \)
5 \( 1 + (-2.18 - 0.467i)T \)
7 \( 1 + (-2.41 - 1.07i)T \)
good11 \( 1 + (-2.98 + 5.16i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (5.24 + 3.02i)T + (6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.10 + 0.636i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.15 + 5.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.86 - 1.07i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.526 - 0.911i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 7.24T + 31T^{2} \)
37 \( 1 + (-1.76 - 1.01i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (4.93 - 8.54i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-5.14 + 2.96i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + 0.936iT - 47T^{2} \)
53 \( 1 + (-2.66 + 1.53i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 8.72T + 59T^{2} \)
61 \( 1 + 1.07T + 61T^{2} \)
67 \( 1 - 0.244iT - 67T^{2} \)
71 \( 1 + 12.6T + 71T^{2} \)
73 \( 1 + (6.08 - 3.51i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 - 11.0T + 79T^{2} \)
83 \( 1 + (-1.40 + 0.810i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (5.31 - 9.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (11.3 - 6.52i)T + (48.5 - 84.0i)T^{2} \)
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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

−10.63063571379148816401502150710, −9.747787748628485158075699928054, −9.026891680064390210023667786509, −7.922191847410453920709946858376, −6.89101776680911849102307965756, −6.01777941509547873074982548627, −5.35395282588794851344824346944, −4.66358366439839091642092694854, −2.92864151455562044512941195402, −1.01252489503554263596243397818, 1.41561962050108790464527330188, 2.09559644303537042405179341683, 4.33173563870902411857315139083, 4.76053849290175898198544835006, 5.83400307244653333052412948362, 6.95963556311254501227875876469, 7.72036775269808106628873647873, 9.195534380419132710252357638967, 10.04530410278595743491194063481, 10.25749550812306598347347899079

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