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Properties

Label 2-1638-13.9-c1-0-31
Degree $2$
Conductor $1638$
Sign $-0.128 + 0.991i$
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 + 3.56·5-s + (0.5 + 0.866i)7-s − 0.999·8-s + (1.78 − 3.08i)10-s + (1.28 − 2.21i)11-s + (−0.5 − 3.57i)13-s + 0.999·14-s + (−0.5 + 0.866i)16-s + (−2.5 − 4.33i)17-s + (−3.84 − 6.65i)19-s + (−1.78 − 3.08i)20-s + (−1.28 − 2.21i)22-s + (−4.56 + 7.90i)23-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + 1.59·5-s + (0.188 + 0.327i)7-s − 0.353·8-s + (0.563 − 0.975i)10-s + (0.386 − 0.668i)11-s + (−0.138 − 0.990i)13-s + 0.267·14-s + (−0.125 + 0.216i)16-s + (−0.606 − 1.05i)17-s + (−0.881 − 1.52i)19-s + (−0.398 − 0.689i)20-s + (−0.273 − 0.472i)22-s + (−0.951 + 1.64i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.128 + 0.991i$
Analytic conductor: \(13.0794\)
Root analytic conductor: \(3.61655\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1638} (1387, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1638,\ (\ :1/2),\ -0.128 + 0.991i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.589733796\)
\(L(\frac12)\) \(\approx\) \(2.589733796\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.56T + 5T^{2} \)
11 \( 1 + (-1.28 + 2.21i)T + (-5.5 - 9.52i)T^{2} \)
17 \( 1 + (2.5 + 4.33i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.84 + 6.65i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (4.56 - 7.90i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 5.12T + 31T^{2} \)
37 \( 1 + (-4.90 + 8.49i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (-2.06 + 3.57i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.43 - 2.49i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 - 7.68T + 47T^{2} \)
53 \( 1 - 3.87T + 53T^{2} \)
59 \( 1 + (-6.56 - 11.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.561 - 0.972i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (2 + 3.46i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 2.43T + 73T^{2} \)
79 \( 1 - 1.43T + 79T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (4.84 - 8.38i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.12 + 10.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.210489854934912527635228253443, −8.832939273795010566951530774731, −7.52203502932342877292761708014, −6.45812639877375586890079077757, −5.70446155017443509489202238294, −5.22763120971291100771364331775, −4.10347308173524571284447496554, −2.73548747532475652203180691112, −2.27151998198917873306010280366, −0.889979615086123840574658411657, 1.67487370351968698523625197531, 2.38622936342716217461823572677, 4.11461606066529193568456421785, 4.52415974782371186942518617300, 5.79266668701004165674870396042, 6.37016461487945151813995058249, 6.78990101602808139368808337797, 8.132385509010578259128154516716, 8.647237322383905121617576613203, 9.740502573184283576605687212575

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