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Properties

Label 2-1638-273.68-c1-0-25
Degree $2$
Conductor $1638$
Sign $0.313 + 0.949i$
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 + (−1.29 + 2.23i)5-s + (−1.70 + 2.02i)7-s − i·8-s + (−2.23 − 1.29i)10-s + (−3.04 − 1.75i)11-s + (2.97 + 2.03i)13-s + (−2.02 − 1.70i)14-s + 16-s − 1.71·17-s + (−4.49 + 2.59i)19-s + (1.29 − 2.23i)20-s + (1.75 − 3.04i)22-s − 5.34i·23-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.5·4-s + (−0.577 + 1.00i)5-s + (−0.644 + 0.764i)7-s − 0.353i·8-s + (−0.707 − 0.408i)10-s + (−0.918 − 0.530i)11-s + (0.824 + 0.565i)13-s + (−0.540 − 0.455i)14-s + 0.250·16-s − 0.416·17-s + (−1.03 + 0.595i)19-s + (0.288 − 0.500i)20-s + (0.375 − 0.649i)22-s − 1.11i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.313 + 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.313 + 0.949i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.313 + 0.949i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.01821996354\)
\(L(\frac12)\) \(\approx\) \(0.01821996354\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.70 - 2.02i)T \)
13 \( 1 + (-2.97 - 2.03i)T \)
good5 \( 1 + (1.29 - 2.23i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (3.04 + 1.75i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 + 1.71T + 17T^{2} \)
19 \( 1 + (4.49 - 2.59i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + 5.34iT - 23T^{2} \)
29 \( 1 + (-3.33 + 1.92i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (8.51 - 4.91i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 8.94T + 37T^{2} \)
41 \( 1 + (3.84 + 6.65i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.447 + 0.775i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.287 + 0.497i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.854 - 0.493i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.41T + 59T^{2} \)
61 \( 1 + (-11.7 + 6.76i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.95 + 3.39i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.92 + 2.84i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.53 - 2.61i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.410 + 0.710i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 5.14T + 83T^{2} \)
89 \( 1 + 6.15T + 89T^{2} \)
97 \( 1 + (-8.60 - 4.97i)T + (48.5 + 84.0i)T^{2} \)
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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

−8.870717977103464020472072942539, −8.489203180135428758564128762876, −7.57745425909224249183246053682, −6.63382807732500812599661425746, −6.25423444291467189403424356908, −5.32425542660465462633027224388, −4.11908502740976654517875510349, −3.29484569339667881633896072049, −2.30006719201514555095150485570, −0.00778832371060795632300350455, 1.13580812453971810224028775486, 2.56619257819116385791641552914, 3.69851937370622923859494309065, 4.36741467352899410248758875662, 5.18517596985687055220781022117, 6.23629332207565528794899320826, 7.37999319360144216067808419851, 8.062388043147770874125605502620, 8.826926307309476355654074980266, 9.584152209383416010694956081673

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