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Properties

Label 2-1638-21.5-c1-0-15
Degree $2$
Conductor $1638$
Sign $0.999 - 0.0294i$
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 + (−0.954 + 1.65i)5-s + (1.84 + 1.89i)7-s − 0.999i·8-s + (1.65 − 0.954i)10-s + (4.01 − 2.31i)11-s − i·13-s + (−0.649 − 2.56i)14-s + (−0.5 + 0.866i)16-s + (−2.91 − 5.04i)17-s + (3.60 + 2.08i)19-s − 1.90·20-s − 4.63·22-s + (3.66 + 2.11i)23-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.426 + 0.739i)5-s + (0.697 + 0.716i)7-s − 0.353i·8-s + (0.522 − 0.301i)10-s + (1.20 − 0.698i)11-s − 0.277i·13-s + (−0.173 − 0.685i)14-s + (−0.125 + 0.216i)16-s + (−0.706 − 1.22i)17-s + (0.827 + 0.477i)19-s − 0.426·20-s − 0.987·22-s + (0.765 + 0.441i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.999 - 0.0294i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.999 - 0.0294i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.382874985\)
\(L(\frac12)\) \(\approx\) \(1.382874985\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.84 - 1.89i)T \)
13 \( 1 + iT \)
good5 \( 1 + (0.954 - 1.65i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-4.01 + 2.31i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.91 + 5.04i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.60 - 2.08i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-3.66 - 2.11i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 8.86iT - 29T^{2} \)
31 \( 1 + (0.0915 - 0.0528i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-5.09 + 8.81i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 1.70T + 43T^{2} \)
47 \( 1 + (-0.0414 + 0.0717i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.80 - 3.35i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.74 - 8.22i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-10.6 - 6.16i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.35 - 5.81i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 6.63iT - 71T^{2} \)
73 \( 1 + (1.42 - 0.820i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.21 + 10.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 2.25T + 83T^{2} \)
89 \( 1 + (-6.11 + 10.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.369iT - 97T^{2} \)
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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

−9.221788114959075957967116501806, −8.810823949149660940813888295118, −7.77619901948269522717917685922, −7.23189530260829381739339683959, −6.27552160140497966084899852488, −5.35917060003805049181685129016, −4.14503018513500513969433391381, −3.20428987859028809749279703876, −2.32630576497341981170605178837, −0.943427427038876556504426388429, 0.966321092301392581436332235075, 1.79964948250866413058665287932, 3.57630857607810812343630509302, 4.58057771634911358033992444731, 5.04411948801747372870448140778, 6.61811242987255054027945506293, 6.85731835492736656763724201820, 7.989600863020964195843159923393, 8.516589551144974986877855524363, 9.225157072123150260584837558371

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