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Properties

Label 2-1638-13.10-c1-0-19
Degree $2$
Conductor $1638$
Sign $0.996 - 0.0797i$
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 + 3.18i·5-s + (0.866 + 0.5i)7-s − 0.999i·8-s + (1.59 + 2.75i)10-s + (3.49 − 2.01i)11-s + (−0.333 − 3.59i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (1.23 − 2.13i)17-s + (0.595 + 0.343i)19-s + (2.75 + 1.59i)20-s + (2.01 − 3.49i)22-s + (2.89 + 5.01i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 1.42i·5-s + (0.327 + 0.188i)7-s − 0.353i·8-s + (0.503 + 0.871i)10-s + (1.05 − 0.608i)11-s + (−0.0925 − 0.995i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (0.298 − 0.517i)17-s + (0.136 + 0.0788i)19-s + (0.616 + 0.355i)20-s + (0.429 − 0.744i)22-s + (0.603 + 1.04i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.996 - 0.0797i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.996 - 0.0797i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.776393405\)
\(L(\frac12)\) \(\approx\) \(2.776393405\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.333 + 3.59i)T \)
good5 \( 1 - 3.18iT - 5T^{2} \)
11 \( 1 + (-3.49 + 2.01i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.23 + 2.13i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.595 - 0.343i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.89 - 5.01i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.940 - 1.62i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 4.17iT - 31T^{2} \)
37 \( 1 + (0.322 - 0.186i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6.83 + 3.94i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.48 + 4.30i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 11.8iT - 47T^{2} \)
53 \( 1 - 9.59T + 53T^{2} \)
59 \( 1 + (-2.23 - 1.28i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (6.79 - 11.7i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.46 + 4.30i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (9.08 + 5.24i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 16.5iT - 73T^{2} \)
79 \( 1 + 13.2T + 79T^{2} \)
83 \( 1 + 9.39iT - 83T^{2} \)
89 \( 1 + (-7.55 + 4.36i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.32 + 3.07i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.542151954648198413821483467127, −8.687364622808177416767319622277, −7.47427107870414629557402156905, −7.02166407489691021968660557827, −5.98113535034032303715034547849, −5.45644431816364556029486624353, −4.17108696034467267694017045193, −3.22598257254841351757621455416, −2.72639865812474435270555542153, −1.23604589551708631757880039394, 1.10004135052111374271776522232, 2.19634750366983714545599702794, 3.87236000021475097725937572961, 4.43221314079495573877756051133, 5.05706809357876312300273059686, 6.10191277539369852629920919059, 6.86037232567704664410831981590, 7.76890683837375474907772867511, 8.628335844473711346460086978250, 9.151350746795450580062187841449

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