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Properties

Label 2-3344-836.151-c0-0-3
Degree $2$
Conductor $3344$
Sign $0.970 + 0.242i$
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 + (0.690 − 0.951i)17-s + (−0.309 − 0.951i)19-s + 1.17i·23-s + (−0.190 − 0.587i)25-s + (0.309 − 0.224i)35-s − 1.61·43-s + 0.618·45-s + (1.11 − 0.363i)47-s + (0.5 + 0.363i)49-s + (−0.190 + 0.587i)55-s + (−0.690 + 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 + (0.690 − 0.951i)17-s + (−0.309 − 0.951i)19-s + 1.17i·23-s + (−0.190 − 0.587i)25-s + (0.309 − 0.224i)35-s − 1.61·43-s + 0.618·45-s + (1.11 − 0.363i)47-s + (0.5 + 0.363i)49-s + (−0.190 + 0.587i)55-s + (−0.690 + 0.951i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3344\)    =    \(2^{4} \cdot 11 \cdot 19\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(1.66887\)
Root analytic conductor: \(1.29184\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3344} (1823, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3344,\ (\ :0),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.559704587\)
\(L(\frac12)\) \(\approx\) \(1.559704587\)
\(L(1)\) 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 \( 1 + (-0.690 + 0.951i)T + (-0.309 - 0.951i)T^{2} \)
23 \( 1 - 1.17iT - T^{2} \)
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 \( 1 + (-1.11 + 0.363i)T + (0.809 - 0.587i)T^{2} \)
53 \( 1 + (-0.309 + 0.951i)T^{2} \)
59 \( 1 + (-0.809 - 0.587i)T^{2} \)
61 \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + (0.309 + 0.951i)T^{2} \)
73 \( 1 + (0.809 + 0.587i)T^{2} \)
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

−9.000016516072289208060105339419, −7.81120180564404131775635519283, −7.13977479407885656366882930744, −6.78934078975870970556388267857, −5.80214186867966122525186773838, −4.84673432837624027623099374831, −4.18137510394533177172483210539, −3.25218948973328309592344041073, −2.16177919694784782703225235976, −1.12627357197810526454179804593, 1.35250255096819941793850799497, 2.11278064741072929724577942664, 3.34308004457439547300499841212, 4.20009737922579577558355625801, 5.13697470992198845414839055568, 5.81995431923891140718471889795, 6.40653188513065214287274127318, 7.44323314375693585933078446853, 8.295842105971668263192385222957, 8.641118138292268447730288981648

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