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Properties

Label 2-1638-91.34-c1-0-17
Degree $2$
Conductor $1638$
Sign $-0.0464 - 0.998i$
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 + (−2.27 + 2.27i)5-s + (1.35 + 2.27i)7-s + (0.707 − 0.707i)8-s + 3.21·10-s + (−0.355 + 0.355i)11-s + (2 + 3i)13-s + (0.648 − 2.56i)14-s − 1.00·16-s + 4.32·17-s + (5.98 − 5.98i)19-s + (−2.27 − 2.27i)20-s + 0.502·22-s + 2.38i·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−1.01 + 1.01i)5-s + (0.512 + 0.858i)7-s + (0.250 − 0.250i)8-s + 1.01·10-s + (−0.107 + 0.107i)11-s + (0.554 + 0.832i)13-s + (0.173 − 0.685i)14-s − 0.250·16-s + 1.04·17-s + (1.37 − 1.37i)19-s + (−0.508 − 0.508i)20-s + 0.107·22-s + 0.497i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0464 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0464 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.0464 - 0.998i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.0464 - 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.043319745\)
\(L(\frac12)\) \(\approx\) \(1.043319745\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.35 - 2.27i)T \)
13 \( 1 + (-2 - 3i)T \)
good5 \( 1 + (2.27 - 2.27i)T - 5iT^{2} \)
11 \( 1 + (0.355 - 0.355i)T - 11iT^{2} \)
17 \( 1 - 4.32T + 17T^{2} \)
19 \( 1 + (-5.98 + 5.98i)T - 19iT^{2} \)
23 \( 1 - 2.38iT - 23T^{2} \)
29 \( 1 + 1.09T + 29T^{2} \)
31 \( 1 + (-1.08 + 1.08i)T - 31iT^{2} \)
37 \( 1 + (5.18 - 5.18i)T - 37iT^{2} \)
41 \( 1 + (3.53 - 3.53i)T - 41iT^{2} \)
43 \( 1 - 7.44iT - 43T^{2} \)
47 \( 1 + (-4.71 - 4.71i)T + 47iT^{2} \)
53 \( 1 - 11.2T + 53T^{2} \)
59 \( 1 + (3.61 + 3.61i)T + 59iT^{2} \)
61 \( 1 - 4.32iT - 61T^{2} \)
67 \( 1 + (0.531 + 0.531i)T + 67iT^{2} \)
71 \( 1 + (6.38 + 6.38i)T + 71iT^{2} \)
73 \( 1 + (5.18 + 5.18i)T + 73iT^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + (6.71 - 6.71i)T - 83iT^{2} \)
89 \( 1 + (-3.75 - 3.75i)T + 89iT^{2} \)
97 \( 1 + (12.9 - 12.9i)T - 97iT^{2} \)
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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

−9.523722133245956834627998181554, −8.870398782554219463221828038162, −7.953746768027969033123959361357, −7.41292879670366625065643597940, −6.62357322058445300262622723347, −5.45856912573044498867303372294, −4.42481257189498487059446709072, −3.32334642627455137769110034897, −2.75354948275856921950342256404, −1.35435425111504765275977222094, 0.56468338138768667919749442252, 1.41201608414328391874886411718, 3.44691996644152518973246051846, 4.11772369838688487099028472964, 5.26794666224752278553807228665, 5.70621565850708391166521113454, 7.28760592608175622963533479087, 7.55368630023899234555404576951, 8.403103938009878931686442768851, 8.769982852802658851246681148012

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