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Properties

Label 2-1638-91.34-c1-0-12
Degree $2$
Conductor $1638$
Sign $-0.924 - 0.381i$
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  + (0.707 + 0.707i)2-s + 1.00i·4-s + (0.461 − 0.461i)5-s + (−0.292 + 2.62i)7-s + (−0.707 + 0.707i)8-s + 0.652·10-s + (−1.60 + 1.60i)11-s + (−2.51 + 2.58i)13-s + (−2.06 + 1.65i)14-s − 1.00·16-s + 6.40·17-s + (−2.52 + 2.52i)19-s + (0.461 + 0.461i)20-s − 2.27·22-s − 8.51i·23-s + ⋯
L(s)  = 1  + (0.499 + 0.499i)2-s + 0.500i·4-s + (0.206 − 0.206i)5-s + (−0.110 + 0.993i)7-s + (−0.250 + 0.250i)8-s + 0.206·10-s + (−0.483 + 0.483i)11-s + (−0.697 + 0.716i)13-s + (−0.552 + 0.441i)14-s − 0.250·16-s + 1.55·17-s + (−0.579 + 0.579i)19-s + (0.103 + 0.103i)20-s − 0.483·22-s − 1.77i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 - 0.381i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.924 - 0.381i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.924 - 0.381i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.537903962\)
\(L(\frac12)\) \(\approx\) \(1.537903962\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 - 0.707i)T \)
3 \( 1 \)
7 \( 1 + (0.292 - 2.62i)T \)
13 \( 1 + (2.51 - 2.58i)T \)
good5 \( 1 + (-0.461 + 0.461i)T - 5iT^{2} \)
11 \( 1 + (1.60 - 1.60i)T - 11iT^{2} \)
17 \( 1 - 6.40T + 17T^{2} \)
19 \( 1 + (2.52 - 2.52i)T - 19iT^{2} \)
23 \( 1 + 8.51iT - 23T^{2} \)
29 \( 1 + 7.91T + 29T^{2} \)
31 \( 1 + (0.922 - 0.922i)T - 31iT^{2} \)
37 \( 1 + (0.953 - 0.953i)T - 37iT^{2} \)
41 \( 1 + (3.89 - 3.89i)T - 41iT^{2} \)
43 \( 1 + 4.17iT - 43T^{2} \)
47 \( 1 + (-1.41 - 1.41i)T + 47iT^{2} \)
53 \( 1 + 9.10T + 53T^{2} \)
59 \( 1 + (-1.17 - 1.17i)T + 59iT^{2} \)
61 \( 1 - 14.4iT - 61T^{2} \)
67 \( 1 + (-3.47 - 3.47i)T + 67iT^{2} \)
71 \( 1 + (-8.29 - 8.29i)T + 71iT^{2} \)
73 \( 1 + (-5.89 - 5.89i)T + 73iT^{2} \)
79 \( 1 + 9.18T + 79T^{2} \)
83 \( 1 + (-9.98 + 9.98i)T - 83iT^{2} \)
89 \( 1 + (-8.54 - 8.54i)T + 89iT^{2} \)
97 \( 1 + (0.272 - 0.272i)T - 97iT^{2} \)
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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

−9.620252666684370514291930327510, −8.863672406183450032389244122635, −8.069858464353615575397157685659, −7.29154813305626060244183247370, −6.41407347160575673968874617957, −5.54141991321101292880940599816, −5.02718287148729603896726520015, −3.96985697633572757985399391032, −2.82920247753287371136783692925, −1.87168502620769290178333505402, 0.46864015806843788458586799419, 1.87726431410801738908441359651, 3.17391548181708714875860211826, 3.68589831173953989189671296246, 4.95330548974823555832834087433, 5.56055053758931070431444838424, 6.51472271722701100957091163366, 7.56583154090075288374342184673, 7.966251106722703410088093356350, 9.432488933832091944801901215735

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