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Properties

Label 2-3344-209.37-c0-0-1
Degree $2$
Conductor $3344$
Sign $0.999 - 0.0237i$
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  + (−0.5 + 0.363i)5-s + (−0.190 − 0.587i)7-s + (−0.809 − 0.587i)9-s + (−0.309 + 0.951i)11-s + (1.30 − 0.951i)17-s + (−0.309 + 0.951i)19-s + 1.61·23-s + (−0.190 + 0.587i)25-s + (0.309 + 0.224i)35-s + 1.61·43-s + 0.618·45-s + (0.5 − 1.53i)47-s + (0.5 − 0.363i)49-s + (−0.190 − 0.587i)55-s + (1.30 − 0.951i)61-s + ⋯
L(s)  = 1  + (−0.5 + 0.363i)5-s + (−0.190 − 0.587i)7-s + (−0.809 − 0.587i)9-s + (−0.309 + 0.951i)11-s + (1.30 − 0.951i)17-s + (−0.309 + 0.951i)19-s + 1.61·23-s + (−0.190 + 0.587i)25-s + (0.309 + 0.224i)35-s + 1.61·43-s + 0.618·45-s + (0.5 − 1.53i)47-s + (0.5 − 0.363i)49-s + (−0.190 − 0.587i)55-s + (1.30 − 0.951i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.999 - 0.0237i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (2545, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.999 - 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.054517276\)
\(L(\frac12)\) \(\approx\) \(1.054517276\)
\(L(1)\) 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 \( 1 + (0.309 - 0.951i)T \)
19 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (0.809 + 0.587i)T^{2} \)
5 \( 1 + (0.5 - 0.363i)T + (0.309 - 0.951i)T^{2} \)
7 \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \)
13 \( 1 + (-0.309 - 0.951i)T^{2} \)
17 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
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 \( 1 - 1.61T + T^{2} \)
47 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 - 0.951i)T^{2} \)
59 \( 1 + (0.809 - 0.587i)T^{2} \)
61 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (-0.618 - 1.90i)T + (-0.809 + 0.587i)T^{2} \)
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

−8.815833550357306047362067439243, −7.923145455187183599154427910476, −7.25507137657271529100270568531, −6.83699722008013936751361593403, −5.68072386899814246690698001449, −5.10475348357503112642005998146, −3.93561726176534389184590118588, −3.36859988836440591949396198420, −2.44753440973454306860509336759, −0.913152735239857571485439198883, 0.907109641643041478268658750132, 2.51569615785464523143033526680, 3.10310809021227902997634672197, 4.14865702750431453499248881730, 5.14096346856855665184261865038, 5.70426849167449092269378511321, 6.40156592824352019884121558514, 7.58514387048734946451662978092, 8.060667575877180004425394284060, 8.841954887192166689641510269993

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