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Properties

Label 2-1638-13.4-c1-0-27
Degree $2$
Conductor $1638$
Sign $-0.636 + 0.770i$
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 + 3.38i·5-s + (0.866 − 0.5i)7-s − 0.999i·8-s + (1.69 − 2.93i)10-s + (−0.712 − 0.411i)11-s + (−2.74 − 2.33i)13-s − 0.999·14-s + (−0.5 + 0.866i)16-s + (−2.29 − 3.96i)17-s + (−5.11 + 2.95i)19-s + (−2.93 + 1.69i)20-s + (0.411 + 0.712i)22-s + (3.06 − 5.30i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + 1.51i·5-s + (0.327 − 0.188i)7-s − 0.353i·8-s + (0.535 − 0.928i)10-s + (−0.214 − 0.124i)11-s + (−0.762 − 0.646i)13-s − 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.555 − 0.962i)17-s + (−1.17 + 0.677i)19-s + (−0.656 + 0.378i)20-s + (0.0877 + 0.151i)22-s + (0.639 − 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.636 + 0.770i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.636 + 0.770i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.636 + 0.770i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3489801837\)
\(L(\frac12)\) \(\approx\) \(0.3489801837\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.74 + 2.33i)T \)
good5 \( 1 - 3.38iT - 5T^{2} \)
11 \( 1 + (0.712 + 0.411i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.29 + 3.96i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (5.11 - 2.95i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3.06 + 5.30i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (3.43 - 5.94i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + 4.28iT - 31T^{2} \)
37 \( 1 + (8.39 + 4.84i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-0.0774 - 0.0446i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.67 - 6.36i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 11.1iT - 47T^{2} \)
53 \( 1 - 7.01T + 53T^{2} \)
59 \( 1 + (1.50 - 0.870i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.18 + 2.06i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.252 - 0.145i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (9.48 - 5.47i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + 12.7iT - 73T^{2} \)
79 \( 1 - 9.95T + 79T^{2} \)
83 \( 1 + 3.23iT - 83T^{2} \)
89 \( 1 + (6.96 + 4.02i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (-12.7 + 7.38i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.128687651334172995174709709578, −8.335642753472153944179758535003, −7.34079684565019894387880705732, −7.03573285839534686488053694154, −6.06486770200072431270940743928, −4.90507197307536261396958564945, −3.72506308943262425294860255855, −2.78713441193942321906303362060, −2.09592991444404758346045938817, −0.16023752689218488783500413660, 1.37158764093879750724689664774, 2.28012032687777358265253611413, 4.07724472570487124447950129454, 4.83718730072515938933287147319, 5.51722177001712952959365065906, 6.53710817701216606885653291950, 7.44063697021978196097865065754, 8.269808848678991394403408977170, 8.909919420582610003889286675684, 9.292806056201695341897659910878

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