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Properties

Label 2-1638-13.3-c1-0-27
Degree $2$
Conductor $1638$
Sign $0.396 + 0.918i$
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  + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s − 0.561·5-s + (0.5 − 0.866i)7-s − 0.999·8-s + (−0.280 − 0.486i)10-s + (−0.780 − 1.35i)11-s + (−0.5 − 3.57i)13-s + 0.999·14-s + (−0.5 − 0.866i)16-s + (−2.5 + 4.33i)17-s + (2.34 − 4.05i)19-s + (0.280 − 0.486i)20-s + (0.780 − 1.35i)22-s + (−0.438 − 0.759i)23-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s − 0.251·5-s + (0.188 − 0.327i)7-s − 0.353·8-s + (−0.0887 − 0.153i)10-s + (−0.235 − 0.407i)11-s + (−0.138 − 0.990i)13-s + 0.267·14-s + (−0.125 − 0.216i)16-s + (−0.606 + 1.05i)17-s + (0.537 − 0.930i)19-s + (0.0627 − 0.108i)20-s + (0.166 − 0.288i)22-s + (−0.0914 − 0.158i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.396 + 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $0.396 + 0.918i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (757, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ 0.396 + 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.127178114\)
\(L(\frac12)\) \(\approx\) \(1.127178114\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 \)
7 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.5 + 3.57i)T \)
good5 \( 1 + 0.561T + 5T^{2} \)
11 \( 1 + (0.780 + 1.35i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (2.5 - 4.33i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.34 + 4.05i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.438 + 0.759i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 3.12T + 31T^{2} \)
37 \( 1 + (5.40 + 9.35i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.06 + 3.57i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-5.56 + 9.63i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 4.68T + 47T^{2} \)
53 \( 1 - 12.1T + 53T^{2} \)
59 \( 1 + (-2.43 + 4.22i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.56 - 6.16i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (2 - 3.46i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + 6.56T + 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + 6.24T + 83T^{2} \)
89 \( 1 + (-1.34 - 2.32i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-2.12 + 3.67i)T + (-48.5 - 84.0i)T^{2} \)
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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

−8.931312959522777605540615231633, −8.388465471171635172911955556786, −7.50079899508412070730716770311, −6.98366817412523824049150000269, −5.80344697089469087292295290330, −5.33347004721397274062283651296, −4.16517297207038882530900349336, −3.50032755034561216328301168621, −2.22445213071692430015214103934, −0.37620182914749083697958668464, 1.51591817813661463527776482734, 2.49866684084187457364595598453, 3.57909954174109377820174075796, 4.51694200208527565383278998304, 5.20584365592663464243387659177, 6.20939604945035753021468087472, 7.13319837145809456830200718872, 7.957388400860586997503029977417, 8.932648891981961414184192704100, 9.616393914919259206183898814056

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