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Properties

Label 2-3344-209.113-c0-0-0
Degree $2$
Conductor $3344$
Sign $-0.479 - 0.877i$
Analytic cond. $1.66887$
Root an. cond. $1.29184$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.58 + 1.14i)5-s + (−0.413 + 1.27i)7-s + (−0.809 + 0.587i)9-s + (−0.669 + 0.743i)11-s + (−1.47 − 1.07i)17-s + (−0.309 − 0.951i)19-s + 1.61·23-s + (0.873 + 2.68i)25-s + (−2.11 + 1.53i)35-s + 0.209·43-s − 1.95·45-s + (0.0646 + 0.198i)47-s + (−0.639 − 0.464i)49-s + (−1.91 + 0.406i)55-s + (0.169 + 0.122i)61-s + ⋯
L(s)  = 1  + (1.58 + 1.14i)5-s + (−0.413 + 1.27i)7-s + (−0.809 + 0.587i)9-s + (−0.669 + 0.743i)11-s + (−1.47 − 1.07i)17-s + (−0.309 − 0.951i)19-s + 1.61·23-s + (0.873 + 2.68i)25-s + (−2.11 + 1.53i)35-s + 0.209·43-s − 1.95·45-s + (0.0646 + 0.198i)47-s + (−0.639 − 0.464i)49-s + (−1.91 + 0.406i)55-s + (0.169 + 0.122i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.479 - 0.877i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (113, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ -0.479 - 0.877i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.275969427\)
\(L(\frac12)\) \(\approx\) \(1.275969427\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (0.809 - 0.587i)T^{2} \)
5 \( 1 + (-1.58 - 1.14i)T + (0.309 + 0.951i)T^{2} \)
7 \( 1 + (0.413 - 1.27i)T + (-0.809 - 0.587i)T^{2} \)
13 \( 1 + (-0.309 + 0.951i)T^{2} \)
17 \( 1 + (1.47 + 1.07i)T + (0.309 + 0.951i)T^{2} \)
23 \( 1 - 1.61T + T^{2} \)
29 \( 1 + (0.809 + 0.587i)T^{2} \)
31 \( 1 + (-0.309 + 0.951i)T^{2} \)
37 \( 1 + (0.809 + 0.587i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 - 0.209T + T^{2} \)
47 \( 1 + (-0.0646 - 0.198i)T + (-0.809 + 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (0.809 + 0.587i)T^{2} \)
61 \( 1 + (-0.169 - 0.122i)T + (0.309 + 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.309 - 0.951i)T + (-0.809 - 0.587i)T^{2} \)
79 \( 1 + (-0.309 + 0.951i)T^{2} \)
83 \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.309 + 0.951i)T^{2} \)
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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

−9.202243063460902253597869576461, −8.583148044008548032903930142228, −7.24419260628998080546133504629, −6.79284631312826339341559086613, −6.04673708764109857416908576283, −5.30365727230027360139275189329, −4.84264316476964723652985618673, −2.90712465344457956509136899414, −2.62715049272028536884273396896, −2.08463150631497355940375962182, 0.70597611889159737030225661894, 1.78964576931872483813959848984, 2.88967321233264231823417393549, 3.93305958075484151937457393960, 4.80656153291367570739118564268, 5.63816409435411985528918950259, 6.22364538283782206696572973138, 6.80933374820291450294857206446, 8.052182037163633557014946766231, 8.747674307957724746648128723993

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