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Properties

Label 2-1452-3.2-c2-0-5
Degree $2$
Conductor $1452$
Sign $-0.716 - 0.697i$
Analytic cond. $39.5641$
Root an. cond. $6.29000$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.14 − 2.09i)3-s + 4.10i·5-s − 9.08·7-s + (0.233 + 8.99i)9-s + 17.0·13-s + (8.59 − 8.81i)15-s + 10.7i·17-s + 9.28·19-s + (19.5 + 19.0i)21-s − 39.8i·23-s + 8.15·25-s + (18.3 − 19.8i)27-s − 16.5i·29-s + 12.0·31-s − 37.2i·35-s + ⋯
L(s)  = 1  + (−0.716 − 0.697i)3-s + 0.820i·5-s − 1.29·7-s + (0.0259 + 0.999i)9-s + 1.31·13-s + (0.572 − 0.587i)15-s + 0.630i·17-s + 0.488·19-s + (0.929 + 0.905i)21-s − 1.73i·23-s + 0.326·25-s + (0.679 − 0.734i)27-s − 0.571i·29-s + 0.390·31-s − 1.06i·35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.716 - 0.697i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1452\)    =    \(2^{2} \cdot 3 \cdot 11^{2}\)
Sign: $-0.716 - 0.697i$
Analytic conductor: \(39.5641\)
Root analytic conductor: \(6.29000\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{1452} (485, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1452,\ (\ :1),\ -0.716 - 0.697i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.4385765569\)
\(L(\frac12)\) \(\approx\) \(0.4385765569\)
\(L(2)\) 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 \( 1 + (2.14 + 2.09i)T \)
11 \( 1 \)
good5 \( 1 - 4.10iT - 25T^{2} \)
7 \( 1 + 9.08T + 49T^{2} \)
13 \( 1 - 17.0T + 169T^{2} \)
17 \( 1 - 10.7iT - 289T^{2} \)
19 \( 1 - 9.28T + 361T^{2} \)
23 \( 1 + 39.8iT - 529T^{2} \)
29 \( 1 + 16.5iT - 841T^{2} \)
31 \( 1 - 12.0T + 961T^{2} \)
37 \( 1 + 26.6T + 1.36e3T^{2} \)
41 \( 1 - 50.1iT - 1.68e3T^{2} \)
43 \( 1 + 77.1T + 1.84e3T^{2} \)
47 \( 1 - 74.9iT - 2.20e3T^{2} \)
53 \( 1 - 18.5iT - 2.80e3T^{2} \)
59 \( 1 - 40.8iT - 3.48e3T^{2} \)
61 \( 1 + 9.45T + 3.72e3T^{2} \)
67 \( 1 - 37.0T + 4.48e3T^{2} \)
71 \( 1 + 102. iT - 5.04e3T^{2} \)
73 \( 1 + 79.5T + 5.32e3T^{2} \)
79 \( 1 + 39.2T + 6.24e3T^{2} \)
83 \( 1 - 92.4iT - 6.88e3T^{2} \)
89 \( 1 - 58.9iT - 7.92e3T^{2} \)
97 \( 1 - 103.T + 9.40e3T^{2} \)
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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

−9.863611678002573917830136262840, −8.715020003189528916069210576169, −7.949994133080703467453549471047, −6.86082619175016310579180733977, −6.39363630709374154544698696005, −5.97695978739094299562276638877, −4.64463813997163896617509475824, −3.43844560078172966516057067083, −2.62788799147715829996720522240, −1.18999151652844131382206657905, 0.15817545215852309582524599452, 1.31423734774577305159796325628, 3.30762785537968549184480871595, 3.73218099712560372306774523745, 5.03271093173498977213383700422, 5.55699531209435575413632083189, 6.47198350274182656485073379304, 7.18511811070823171552113592448, 8.609924471744035442689705829076, 9.073145704958678583070886724189

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