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Properties

Label 2-3344-836.303-c0-0-5
Degree $2$
Conductor $3344$
Sign $0.0219 + 0.999i$
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 − 1.53i)5-s + (1.30 − 0.951i)7-s + (−0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s + (1.80 + 0.587i)17-s + (0.809 + 0.587i)19-s + 1.90i·23-s + (−1.30 − 0.951i)25-s + (−0.809 − 2.48i)35-s + 0.618·43-s − 1.61·45-s + (−1.11 + 1.53i)47-s + (0.500 − 1.53i)49-s + (−1.30 + 0.951i)55-s + (−1.80 − 0.587i)61-s + ⋯
L(s)  = 1  + (0.5 − 1.53i)5-s + (1.30 − 0.951i)7-s + (−0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s + (1.80 + 0.587i)17-s + (0.809 + 0.587i)19-s + 1.90i·23-s + (−1.30 − 0.951i)25-s + (−0.809 − 2.48i)35-s + 0.618·43-s − 1.61·45-s + (−1.11 + 1.53i)47-s + (0.500 − 1.53i)49-s + (−1.30 + 0.951i)55-s + (−1.80 − 0.587i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.0219 + 0.999i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.0219 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.608174138\)
\(L(\frac12)\) \(\approx\) \(1.608174138\)
\(L(1)\) 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 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
good3 \( 1 + (0.309 + 0.951i)T^{2} \)
5 \( 1 + (-0.5 + 1.53i)T + (-0.809 - 0.587i)T^{2} \)
7 \( 1 + (-1.30 + 0.951i)T + (0.309 - 0.951i)T^{2} \)
13 \( 1 + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-1.80 - 0.587i)T + (0.809 + 0.587i)T^{2} \)
23 \( 1 - 1.90iT - T^{2} \)
29 \( 1 + (0.309 - 0.951i)T^{2} \)
31 \( 1 + (-0.809 + 0.587i)T^{2} \)
37 \( 1 + (-0.309 + 0.951i)T^{2} \)
41 \( 1 + (0.309 + 0.951i)T^{2} \)
43 \( 1 - 0.618T + T^{2} \)
47 \( 1 + (1.11 - 1.53i)T + (-0.309 - 0.951i)T^{2} \)
53 \( 1 + (0.809 - 0.587i)T^{2} \)
59 \( 1 + (0.309 - 0.951i)T^{2} \)
61 \( 1 + (1.80 + 0.587i)T + (0.809 + 0.587i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (-0.809 - 0.587i)T^{2} \)
73 \( 1 + (-0.309 + 0.951i)T^{2} \)
79 \( 1 + (0.809 - 0.587i)T^{2} \)
83 \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.809 - 0.587i)T^{2} \)
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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

−8.484638687867476724328472553043, −7.85312849453859328308969784580, −7.57898234594208151457686608284, −6.03078977958096147033337381752, −5.48613497699562779568882564479, −4.98561806253764882482267740202, −3.97155003471512963980302920509, −3.22232970705782257323547847559, −1.41125392094129893698945115263, −1.16919241490393474471383170693, 1.80220877367046157813403250965, 2.61128011374648330496064837222, 3.01076482370695393060820390116, 4.63971465793308962628211733464, 5.27705164949693689067474987495, 5.80386519701732585375891892250, 6.86455403452215269714707672117, 7.62456021127655405913386600095, 8.008367120920119420586741267200, 8.914567616463224968601236995105

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