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Properties

Label 2-1638-91.34-c1-0-36
Degree $2$
Conductor $1638$
Sign $-0.241 + 0.970i$
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 + (2.56 − 2.56i)5-s + (−0.648 − 2.56i)7-s + (0.707 − 0.707i)8-s − 3.62·10-s + (1.64 − 1.64i)11-s + (2 + 3i)13-s + (−1.35 + 2.27i)14-s − 1.00·16-s + 7.15·17-s + (−2.86 + 2.86i)19-s + (2.56 + 2.56i)20-s − 2.33·22-s − 4.45i·23-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + 0.500i·4-s + (1.14 − 1.14i)5-s + (−0.245 − 0.969i)7-s + (0.250 − 0.250i)8-s − 1.14·10-s + (0.496 − 0.496i)11-s + (0.554 + 0.832i)13-s + (−0.362 + 0.607i)14-s − 0.250·16-s + 1.73·17-s + (−0.656 + 0.656i)19-s + (0.573 + 0.573i)20-s − 0.496·22-s − 0.929i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.241 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1638\)    =    \(2 \cdot 3^{2} \cdot 7 \cdot 13\)
Sign: $-0.241 + 0.970i$
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: \((2,\ 1638,\ (\ :1/2),\ -0.241 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.773200918\)
\(L(\frac12)\) \(\approx\) \(1.773200918\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.648 + 2.56i)T \)
13 \( 1 + (-2 - 3i)T \)
good5 \( 1 + (-2.56 + 2.56i)T - 5iT^{2} \)
11 \( 1 + (-1.64 + 1.64i)T - 11iT^{2} \)
17 \( 1 - 7.15T + 17T^{2} \)
19 \( 1 + (2.86 - 2.86i)T - 19iT^{2} \)
23 \( 1 + 4.45iT - 23T^{2} \)
29 \( 1 - 9.75T + 29T^{2} \)
31 \( 1 + (-3.91 + 3.91i)T - 31iT^{2} \)
37 \( 1 + (-2.48 + 2.48i)T - 37iT^{2} \)
41 \( 1 + (7.53 - 7.53i)T - 41iT^{2} \)
43 \( 1 - 8.62iT - 43T^{2} \)
47 \( 1 + (-0.703 - 0.703i)T + 47iT^{2} \)
53 \( 1 + 2.42T + 53T^{2} \)
59 \( 1 + (10.4 + 10.4i)T + 59iT^{2} \)
61 \( 1 - 7.15iT - 61T^{2} \)
67 \( 1 + (4.53 + 4.53i)T + 67iT^{2} \)
71 \( 1 + (-0.456 - 0.456i)T + 71iT^{2} \)
73 \( 1 + (-2.48 - 2.48i)T + 73iT^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + (2.70 - 2.70i)T - 83iT^{2} \)
89 \( 1 + (4.75 + 4.75i)T + 89iT^{2} \)
97 \( 1 + (4.49 - 4.49i)T - 97iT^{2} \)
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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

−9.353771171915455461241537224853, −8.394125136458710488149199885161, −7.980385580733234083739904518631, −6.50182154861164117767861415773, −6.14687137327945816296143946283, −4.83519010924601327283367528608, −4.11225569267437810288602162356, −2.97991403562017103984853117465, −1.52886656984306089881299053865, −0.952224231794907351857241316121, 1.42080425068235545030891416407, 2.59146125104822342816887649837, 3.35873264340533555604377052869, 5.08416950458202816933338074875, 5.77651636891089433752340019786, 6.41239525554318920191616441926, 7.04229239282473412044834723096, 8.072756617404897071442470001537, 8.869195636067037287793690411260, 9.656979283502137282455226781454

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