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Properties

Label 2-3344-76.75-c1-0-94
Degree $2$
Conductor $3344$
Sign $-0.189 + 0.981i$
Analytic cond. $26.7019$
Root an. cond. $5.16739$
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  + 3.25·3-s − 2.27·5-s − 4.28i·7-s + 7.58·9-s + i·11-s − 2.82i·13-s − 7.39·15-s − 5.42·17-s + (4.27 + 0.826i)19-s − 13.9i·21-s − 8.87i·23-s + 0.169·25-s + 14.9·27-s + 5.91i·29-s − 5.48·31-s + ⋯
L(s)  = 1  + 1.87·3-s − 1.01·5-s − 1.61i·7-s + 2.52·9-s + 0.301i·11-s − 0.783i·13-s − 1.90·15-s − 1.31·17-s + (0.981 + 0.189i)19-s − 3.03i·21-s − 1.85i·23-s + 0.0338·25-s + 2.86·27-s + 1.09i·29-s − 0.985·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $-0.189 + 0.981i$
Analytic conductor: \(26.7019\)
Root analytic conductor: \(5.16739\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1519, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :1/2),\ -0.189 + 0.981i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.685684126\)
\(L(\frac12)\) \(\approx\) \(2.685684126\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 - iT \)
19 \( 1 + (-4.27 - 0.826i)T \)
good3 \( 1 - 3.25T + 3T^{2} \)
5 \( 1 + 2.27T + 5T^{2} \)
7 \( 1 + 4.28iT - 7T^{2} \)
13 \( 1 + 2.82iT - 13T^{2} \)
17 \( 1 + 5.42T + 17T^{2} \)
23 \( 1 + 8.87iT - 23T^{2} \)
29 \( 1 - 5.91iT - 29T^{2} \)
31 \( 1 + 5.48T + 31T^{2} \)
37 \( 1 - 3.25iT - 37T^{2} \)
41 \( 1 + 2.92iT - 41T^{2} \)
43 \( 1 + 9.28iT - 43T^{2} \)
47 \( 1 + 5.09iT - 47T^{2} \)
53 \( 1 - 3.16iT - 53T^{2} \)
59 \( 1 + 1.30T + 59T^{2} \)
61 \( 1 - 1.77T + 61T^{2} \)
67 \( 1 - 8.89T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 + 5.70T + 73T^{2} \)
79 \( 1 - 15.6T + 79T^{2} \)
83 \( 1 + 14.9iT - 83T^{2} \)
89 \( 1 - 18.0iT - 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−8.432403449719713641811536128724, −7.63038744313726170535352841407, −7.26618033166879215980465938401, −6.71453650621386235806681444669, −4.92270282325331091137267165951, −4.16141772797712257375228280843, −3.70760820166846777615131942728, −2.98494911634962814557717885093, −1.89406201515623399026991860711, −0.60540288525503456159842599879, 1.68757348078082801766854451239, 2.44980747265005953249505638314, 3.23817799068393917823105219005, 3.91737880444033053741872812659, 4.73691311370985717948406269954, 5.84159207052065047697312454550, 6.90090725103509096564570249203, 7.67133498137140194505169600016, 8.114697188831836846531162904124, 8.850572983869127335857573018916

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