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Properties

Label 2-1638-91.4-c1-0-7
Degree $2$
Conductor $1638$
Sign $-0.999 + 0.0105i$
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  + i·2-s − 4-s + (2.89 − 1.67i)5-s + (0.268 + 2.63i)7-s − i·8-s + (1.67 + 2.89i)10-s + (−4.36 + 2.51i)11-s + (−1.92 + 3.04i)13-s + (−2.63 + 0.268i)14-s + 16-s − 3.73·17-s + (−5.08 − 2.93i)19-s + (−2.89 + 1.67i)20-s + (−2.51 − 4.36i)22-s − 3.78·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (1.29 − 0.747i)5-s + (0.101 + 0.994i)7-s − 0.353i·8-s + (0.528 + 0.915i)10-s + (−1.31 + 0.759i)11-s + (−0.534 + 0.845i)13-s + (−0.703 + 0.0717i)14-s + 0.250·16-s − 0.905·17-s + (−1.16 − 0.673i)19-s + (−0.647 + 0.373i)20-s + (−0.536 − 0.929i)22-s − 0.788·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0105i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0105i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.0105i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.999 + 0.0105i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8388454131\)
\(L(\frac12)\) \(\approx\) \(0.8388454131\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 \)
7 \( 1 + (-0.268 - 2.63i)T \)
13 \( 1 + (1.92 - 3.04i)T \)
good5 \( 1 + (-2.89 + 1.67i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (4.36 - 2.51i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 3.73T + 17T^{2} \)
19 \( 1 + (5.08 + 2.93i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + 3.78T + 23T^{2} \)
29 \( 1 + (3.36 - 5.82i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.884 - 0.510i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.428iT - 37T^{2} \)
41 \( 1 + (0.917 + 0.529i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.65 - 2.85i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.60 + 2.65i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.15 + 2.00i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 + (-5.05 + 8.74i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (11.5 - 6.64i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (2.35 - 1.36i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.88 - 3.97i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.07 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 9.17iT - 83T^{2} \)
89 \( 1 + 5.41iT - 89T^{2} \)
97 \( 1 + (-14.8 + 8.58i)T + (48.5 - 84.0i)T^{2} \)
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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

−9.562531578668273731449756118422, −8.905456568654641531678206707054, −8.422916886861167247965175690752, −7.28148516981999454709897949743, −6.47864958763206881299988622566, −5.64986357945052701745845442046, −5.03462742292499302420380665187, −4.39632426436519404685954629497, −2.45559234265956590040531492023, −1.94206849175731114054303946154, 0.28269469057865518328761252392, 1.99998729437413821619969306774, 2.63212343594646767611592022943, 3.70340545974070490752080315825, 4.77294810924351625628654748911, 5.78286128389676591003155567784, 6.34143265982009221639335980240, 7.55806949754952114302135999678, 8.148393926471587425393975743406, 9.244217384456642427183874897887

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