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Properties

Label 2-1638-13.10-c1-0-2
Degree $2$
Conductor $1638$
Sign $-0.917 + 0.398i$
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 + 1.78i·5-s + (0.866 + 0.5i)7-s + 0.999i·8-s + (−0.894 − 1.54i)10-s + (−2.74 + 1.58i)11-s + (1.47 + 3.28i)13-s − 0.999·14-s + (−0.5 − 0.866i)16-s + (−2.78 + 4.81i)17-s + (−5.36 − 3.09i)19-s + (1.54 + 0.894i)20-s + (1.58 − 2.74i)22-s + (−3.06 − 5.31i)23-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + 0.799i·5-s + (0.327 + 0.188i)7-s + 0.353i·8-s + (−0.282 − 0.489i)10-s + (−0.828 + 0.478i)11-s + (0.410 + 0.911i)13-s − 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.674 + 1.16i)17-s + (−1.22 − 0.709i)19-s + (0.346 + 0.199i)20-s + (0.338 − 0.586i)22-s + (−0.639 − 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.917 + 0.398i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.917 + 0.398i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.917 + 0.398i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.3863967943\)
\(L(\frac12)\) \(\approx\) \(0.3863967943\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.47 - 3.28i)T \)
good5 \( 1 - 1.78iT - 5T^{2} \)
11 \( 1 + (2.74 - 1.58i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.78 - 4.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (5.36 + 3.09i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.06 + 5.31i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.03 - 1.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 5.63iT - 31T^{2} \)
37 \( 1 + (2.68 - 1.54i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-1.29 + 0.749i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.81 - 8.34i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 10.5iT - 47T^{2} \)
53 \( 1 + 3.60T + 53T^{2} \)
59 \( 1 + (-2.40 - 1.39i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.844 - 1.46i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (10.0 - 5.77i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-0.518 - 0.299i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 0.423iT - 73T^{2} \)
79 \( 1 - 6.96T + 79T^{2} \)
83 \( 1 + 4.30iT - 83T^{2} \)
89 \( 1 + (-14.1 + 8.14i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (15.1 + 8.74i)T + (48.5 + 84.0i)T^{2} \)
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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

−9.898255381584870622411602319228, −8.756887900033918379556114256406, −8.420813064309737894212661577753, −7.43657333825593541115096437363, −6.55952878995939993440990722097, −6.22024775102348342323249079711, −4.88781246314802358289169898106, −4.08434053504937993238110540180, −2.59372631858922707868771071369, −1.86602185291687175331336888165, 0.17660463204762279131216189463, 1.43512295046990089292425964970, 2.64980570845063449482869123277, 3.71027137868608999978408402436, 4.81610788494093119935589658693, 5.55253443846906700066765993718, 6.61072904027940879050521293342, 7.71046023384727780576388666425, 8.212339389364268306709096820233, 8.875914563087560943581679200371

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