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Properties

Label 2-1638-13.10-c1-0-22
Degree $2$
Conductor $1638$
Sign $0.311 + 0.950i$
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 + 0.332i·5-s + (−0.866 − 0.5i)7-s − 0.999i·8-s + (0.166 + 0.288i)10-s + (−2.26 + 1.30i)11-s + (3.41 − 1.16i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (1.94 − 3.36i)17-s + (4.85 + 2.80i)19-s + (0.288 + 0.166i)20-s + (−1.30 + 2.26i)22-s + (−2.10 − 3.64i)23-s + ⋯
L(s)  = 1  + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 0.148i·5-s + (−0.327 − 0.188i)7-s − 0.353i·8-s + (0.0526 + 0.0911i)10-s + (−0.681 + 0.393i)11-s + (0.946 − 0.323i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (0.471 − 0.816i)17-s + (1.11 + 0.643i)19-s + (0.0644 + 0.0372i)20-s + (−0.278 + 0.481i)22-s + (−0.439 − 0.760i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.311 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.311 + 0.950i$
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: \((2,\ 1638,\ (\ :1/2),\ 0.311 + 0.950i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.351053428\)
\(L(\frac12)\) \(\approx\) \(2.351053428\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.41 + 1.16i)T \)
good5 \( 1 - 0.332iT - 5T^{2} \)
11 \( 1 + (2.26 - 1.30i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-1.94 + 3.36i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-4.85 - 2.80i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (2.10 + 3.64i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.593 + 1.02i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 7.07iT - 31T^{2} \)
37 \( 1 + (-0.499 + 0.288i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.451 - 0.260i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.53 + 2.66i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 12.0iT - 47T^{2} \)
53 \( 1 - 9.71T + 53T^{2} \)
59 \( 1 + (9.07 + 5.23i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.71 - 6.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.3 + 5.96i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.818 - 0.472i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 4.66iT - 73T^{2} \)
79 \( 1 + 0.943T + 79T^{2} \)
83 \( 1 - 13.7iT - 83T^{2} \)
89 \( 1 + (2.92 - 1.69i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (5.03 + 2.90i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.467122341450245322681825679221, −8.367750895887518300530450217858, −7.54038928009900666139134357339, −6.74940014561860881513918390024, −5.77966528849822350646334928410, −5.15710476354905533571861542277, −4.04969479874292474326173540571, −3.23990902169170392037671910448, −2.30616552449167365414273615638, −0.819170151948762855511968623689, 1.33471838791788171415647508572, 2.89452496836120799673419166325, 3.54162698824933471898446459937, 4.66836609041800098213331958733, 5.53358998092128056145189061409, 6.14765154880555819688879936364, 7.07322499525216528596329389030, 7.88224074426394905918903894761, 8.668825064834070714053365438228, 9.377313244684780833146798927034

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