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Properties

Label 2-1638-13.4-c1-0-21
Degree $2$
Conductor $1638$
Sign $0.994 - 0.107i$
Analytic cond. $13.0794$
Root an. cond. $3.61655$
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  + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s − 1.34i·5-s + (−0.866 + 0.5i)7-s + 0.999i·8-s + (0.674 − 1.16i)10-s + (1.11 + 0.646i)11-s + (−0.343 − 3.58i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (1.76 + 3.05i)17-s + (1.98 − 1.14i)19-s + (1.16 − 0.674i)20-s + (0.646 + 1.11i)22-s + (3.08 − 5.33i)23-s + ⋯
L(s)  = 1  + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s − 0.603i·5-s + (−0.327 + 0.188i)7-s + 0.353i·8-s + (0.213 − 0.369i)10-s + (0.337 + 0.194i)11-s + (−0.0951 − 0.995i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (0.428 + 0.741i)17-s + (0.456 − 0.263i)19-s + (0.261 − 0.150i)20-s + (0.137 + 0.238i)22-s + (0.642 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.994 - 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.994 - 0.107i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.994 - 0.107i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.503715687\)
\(L(\frac12)\) \(\approx\) \(2.503715687\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.866 - 0.5i)T \)
3 \( 1 \)
7 \( 1 + (0.866 - 0.5i)T \)
13 \( 1 + (0.343 + 3.58i)T \)
good5 \( 1 + 1.34iT - 5T^{2} \)
11 \( 1 + (-1.11 - 0.646i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.76 - 3.05i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-1.98 + 1.14i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.08 + 5.33i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.11 + 7.12i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 9.90iT - 31T^{2} \)
37 \( 1 + (-3.71 - 2.14i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-6.84 - 3.95i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (6.26 + 10.8i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 10.7iT - 47T^{2} \)
53 \( 1 - 3.30T + 53T^{2} \)
59 \( 1 + (-9.40 + 5.42i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.64 + 4.57i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.85 + 4.53i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (-6.89 + 3.97i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 3.28iT - 73T^{2} \)
79 \( 1 + 3.32T + 79T^{2} \)
83 \( 1 + 0.731iT - 83T^{2} \)
89 \( 1 + (-11.5 - 6.65i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (10.5 - 6.10i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.259157566064636193817743088682, −8.469373590063317255661270696373, −7.85976490971655123353011995637, −6.80212354696692036433406362722, −6.15224879417571916000835882780, −5.19120814141284909135289127145, −4.59528498747158727413187747030, −3.46919466453715707313448418349, −2.59786690171754385929248088052, −0.985118489785762891483811733156, 1.15929938600890731764833332438, 2.54071656864411538026030260544, 3.36121877616113845859014060555, 4.21641994441871277310287045075, 5.22485058949888922199392118151, 6.08534584405598494616900182394, 6.98240609474884465740650483306, 7.41639673763314657771727243027, 8.758973494537578924078141594808, 9.557942036090105109471542971354

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