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Properties

Label 2-627-11.3-c1-0-32
Degree $2$
Conductor $627$
Sign $0.129 + 0.991i$
Analytic cond. $5.00662$
Root an. cond. $2.23754$
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  + (2.18 − 1.58i)2-s + (0.309 + 0.951i)3-s + (1.64 − 5.04i)4-s + (1.61 + 1.17i)5-s + (2.18 + 1.58i)6-s + (0.601 − 1.85i)7-s + (−2.76 − 8.50i)8-s + (−0.809 + 0.587i)9-s + 5.39·10-s + (−3.27 − 0.531i)11-s + 5.30·12-s + (−2.02 + 1.47i)13-s + (−1.62 − 5.00i)14-s + (−0.617 + 1.89i)15-s + (−10.9 − 7.97i)16-s + (5.13 + 3.73i)17-s + ⋯
L(s)  = 1  + (1.54 − 1.12i)2-s + (0.178 + 0.549i)3-s + (0.820 − 2.52i)4-s + (0.722 + 0.524i)5-s + (0.892 + 0.648i)6-s + (0.227 − 0.699i)7-s + (−0.977 − 3.00i)8-s + (−0.269 + 0.195i)9-s + 1.70·10-s + (−0.987 − 0.160i)11-s + 1.53·12-s + (−0.561 + 0.408i)13-s + (−0.434 − 1.33i)14-s + (−0.159 + 0.490i)15-s + (−2.74 − 1.99i)16-s + (1.24 + 0.905i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 627 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(627\)    =    \(3 \cdot 11 \cdot 19\)
Sign: $0.129 + 0.991i$
Analytic conductor: \(5.00662\)
Root analytic conductor: \(2.23754\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{627} (58, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 627,\ (\ :1/2),\ 0.129 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.88996 - 2.53796i\)
\(L(\frac12)\) \(\approx\) \(2.88996 - 2.53796i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.309 - 0.951i)T \)
11 \( 1 + (3.27 + 0.531i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good2 \( 1 + (-2.18 + 1.58i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + (-1.61 - 1.17i)T + (1.54 + 4.75i)T^{2} \)
7 \( 1 + (-0.601 + 1.85i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.02 - 1.47i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-5.13 - 3.73i)T + (5.25 + 16.1i)T^{2} \)
23 \( 1 - 5.99T + 23T^{2} \)
29 \( 1 + (2.35 - 7.23i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (3.68 - 2.67i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.50 - 7.70i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (2.20 + 6.78i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 - 3.86T + 43T^{2} \)
47 \( 1 + (-0.0432 - 0.133i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (9.42 - 6.84i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-3.10 + 9.55i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (6.33 + 4.60i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 7.21T + 67T^{2} \)
71 \( 1 + (4.09 + 2.97i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.220 - 0.679i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-4.74 + 3.44i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.453 + 0.329i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 6.34T + 89T^{2} \)
97 \( 1 + (9.33 - 6.77i)T + (29.9 - 92.2i)T^{2} \)
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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

−10.55735666805713596704954474136, −10.16419647967976710054747802255, −9.143341359205810762657701710467, −7.48671257000386696076438754439, −6.39603065618971428423954769905, −5.36684459841842371091588569151, −4.79130566921547939295852563162, −3.56694613220943305658671321175, −2.85515254169530659995573413262, −1.62197160920307643695122682065, 2.28682299795357537579484812323, 3.17887457399481158745303388780, 4.74715690622511548020837768867, 5.52318229267157345175590856524, 5.85168082322260983078438903706, 7.30010090812775697875383610770, 7.67327927537481387698687978597, 8.691997559292923567867046464845, 9.709316676944172500960117398718, 11.27361753428157145157642681808

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