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Properties

Label 2-1452-11.4-c1-0-4
Degree $2$
Conductor $1452$
Sign $-0.511 - 0.859i$
Analytic cond. $11.5942$
Root an. cond. $3.40503$
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  + (−0.309 + 0.951i)3-s + (0.535 + 1.64i)7-s + (−0.809 − 0.587i)9-s + (−0.535 + 1.64i)19-s − 1.73·21-s + 6·23-s + (−1.54 + 4.75i)25-s + (0.809 − 0.587i)27-s + (3.21 + 9.88i)29-s + (−5.66 − 4.11i)31-s + (0.309 + 0.951i)37-s + (−3.21 + 9.88i)41-s − 10.3·43-s + (1.85 − 5.70i)47-s + (3.23 − 2.35i)49-s + ⋯
L(s)  = 1  + (−0.178 + 0.549i)3-s + (0.202 + 0.622i)7-s + (−0.269 − 0.195i)9-s + (−0.122 + 0.377i)19-s − 0.377·21-s + 1.25·23-s + (−0.309 + 0.951i)25-s + (0.155 − 0.113i)27-s + (0.596 + 1.83i)29-s + (−1.01 − 0.738i)31-s + (0.0508 + 0.156i)37-s + (−0.501 + 1.54i)41-s − 1.58·43-s + (0.270 − 0.832i)47-s + (0.462 − 0.335i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.511 - 0.859i$
Analytic conductor: \(11.5942\)
Root analytic conductor: \(3.40503\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (565, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1/2),\ -0.511 - 0.859i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.235791695\)
\(L(\frac12)\) \(\approx\) \(1.235791695\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good5 \( 1 + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.535 - 1.64i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.535 - 1.64i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + (-3.21 - 9.88i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (5.66 + 4.11i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-0.309 - 0.951i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (3.21 - 9.88i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + (-1.85 + 5.70i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (4.85 + 3.52i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-1.85 - 5.70i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.40 - 1.01i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 7T + 67T^{2} \)
71 \( 1 + (9.70 - 7.05i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-2.67 - 8.23i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (7.00 + 5.09i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (8.40 - 6.10i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 + (4.04 + 2.93i)T + (29.9 + 92.2i)T^{2} \)
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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

−9.776041299250220748115947898898, −8.956989089832732365751970963334, −8.432406163550210699423923836445, −7.32186728549476403256075116814, −6.50809177138112737111834653375, −5.40933682992091591852980363593, −4.99387154162013202537680584586, −3.75820391297961217739007594831, −2.89502229529087977309379362041, −1.50931879776432531102063450874, 0.51629132975091889551614592442, 1.84683207486561368428746055210, 3.02902199433057051635939375516, 4.20636816841346351610721719455, 5.06156013420018718043648584507, 6.08796410783729952196009717521, 6.87921218528393509667830292152, 7.55818289692231105969088916005, 8.366717997413261063162655965765, 9.171648574445386711687245767891

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