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Properties

Label 2-1472-16.13-c1-0-12
Degree $2$
Conductor $1472$
Sign $-0.590 - 0.806i$
Analytic cond. $11.7539$
Root an. cond. $3.42840$
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  + (1.75 + 1.75i)3-s + (0.238 − 0.238i)5-s + 4.31i·7-s + 3.14i·9-s + (−0.431 + 0.431i)11-s + (−2.08 − 2.08i)13-s + 0.837·15-s + 1.32·17-s + (1.72 + 1.72i)19-s + (−7.56 + 7.56i)21-s + i·23-s + 4.88i·25-s + (−0.262 + 0.262i)27-s + (1.03 + 1.03i)29-s + 2.49·31-s + ⋯
L(s)  = 1  + (1.01 + 1.01i)3-s + (0.106 − 0.106i)5-s + 1.63i·7-s + 1.04i·9-s + (−0.130 + 0.130i)11-s + (−0.579 − 0.579i)13-s + 0.216·15-s + 0.322·17-s + (0.395 + 0.395i)19-s + (−1.65 + 1.65i)21-s + 0.208i·23-s + 0.977i·25-s + (−0.0505 + 0.0505i)27-s + (0.191 + 0.191i)29-s + 0.447·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.806i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1472\)    =    \(2^{6} \cdot 23\)
Sign: $-0.590 - 0.806i$
Analytic conductor: \(11.7539\)
Root analytic conductor: \(3.42840\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1472} (1105, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1472,\ (\ :1/2),\ -0.590 - 0.806i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.195988208\)
\(L(\frac12)\) \(\approx\) \(2.195988208\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
23 \( 1 - iT \)
good3 \( 1 + (-1.75 - 1.75i)T + 3iT^{2} \)
5 \( 1 + (-0.238 + 0.238i)T - 5iT^{2} \)
7 \( 1 - 4.31iT - 7T^{2} \)
11 \( 1 + (0.431 - 0.431i)T - 11iT^{2} \)
13 \( 1 + (2.08 + 2.08i)T + 13iT^{2} \)
17 \( 1 - 1.32T + 17T^{2} \)
19 \( 1 + (-1.72 - 1.72i)T + 19iT^{2} \)
29 \( 1 + (-1.03 - 1.03i)T + 29iT^{2} \)
31 \( 1 - 2.49T + 31T^{2} \)
37 \( 1 + (8.14 - 8.14i)T - 37iT^{2} \)
41 \( 1 + 0.419iT - 41T^{2} \)
43 \( 1 + (-1.04 + 1.04i)T - 43iT^{2} \)
47 \( 1 + 11.5T + 47T^{2} \)
53 \( 1 + (-5.54 + 5.54i)T - 53iT^{2} \)
59 \( 1 + (-5.32 + 5.32i)T - 59iT^{2} \)
61 \( 1 + (7.62 + 7.62i)T + 61iT^{2} \)
67 \( 1 + (-8.86 - 8.86i)T + 67iT^{2} \)
71 \( 1 - 6.26iT - 71T^{2} \)
73 \( 1 + 15.8iT - 73T^{2} \)
79 \( 1 + 5.26T + 79T^{2} \)
83 \( 1 + (-7.33 - 7.33i)T + 83iT^{2} \)
89 \( 1 - 8.04iT - 89T^{2} \)
97 \( 1 - 10.0T + 97T^{2} \)
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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

−9.848192696184366262276796207235, −8.942570174773287805250076266261, −8.459420408674178392789957754239, −7.71040149789434621819642889033, −6.44288666358018859837545713754, −5.28973242688627750270201618673, −4.97036125264661992566246380195, −3.51254253471816372785345622092, −2.95566049960820809950782979066, −1.93418875586443205138949617262, 0.75977095229087800053730657247, 1.93268365466021735138363131156, 2.96419347506929607047054337457, 3.93312043114320887836795147104, 4.87761021402028747052294571052, 6.31699394276020406025341515119, 7.11018967611996116807795290610, 7.47390316655804722642213658806, 8.263612656982579347385843036590, 9.082214415366876281902402135168

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