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Properties

Label 2-344-344.59-c0-0-0
Degree $2$
Conductor $344$
Sign $-0.886 + 0.463i$
Analytic cond. $0.171678$
Root an. cond. $0.414340$
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  + (−0.222 − 0.974i)2-s + (0.400 − 1.75i)3-s + (−0.900 + 0.433i)4-s − 1.80·6-s + (0.623 + 0.781i)8-s + (−2.02 − 0.974i)9-s + (0.400 + 0.193i)11-s + (0.400 + 1.75i)12-s + (0.623 − 0.781i)16-s + (−1.12 + 1.40i)17-s + (−0.500 + 2.19i)18-s + (1.62 − 0.781i)19-s + (0.0990 − 0.433i)22-s + (1.62 − 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (0.400 − 1.75i)3-s + (−0.900 + 0.433i)4-s − 1.80·6-s + (0.623 + 0.781i)8-s + (−2.02 − 0.974i)9-s + (0.400 + 0.193i)11-s + (0.400 + 1.75i)12-s + (0.623 − 0.781i)16-s + (−1.12 + 1.40i)17-s + (−0.500 + 2.19i)18-s + (1.62 − 0.781i)19-s + (0.0990 − 0.433i)22-s + (1.62 − 0.781i)24-s + (−0.222 + 0.974i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $-0.886 + 0.463i$
Analytic conductor: \(0.171678\)
Root analytic conductor: \(0.414340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (59, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 344,\ (\ :0),\ -0.886 + 0.463i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7370580700\)
\(L(\frac12)\) \(\approx\) \(0.7370580700\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 + 0.974i)T \)
43 \( 1 + (0.222 + 0.974i)T \)
good3 \( 1 + (-0.400 + 1.75i)T + (-0.900 - 0.433i)T^{2} \)
5 \( 1 + (0.222 - 0.974i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
13 \( 1 + (0.222 - 0.974i)T^{2} \)
17 \( 1 + (1.12 - 1.40i)T + (-0.222 - 0.974i)T^{2} \)
19 \( 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2} \)
23 \( 1 + (-0.623 - 0.781i)T^{2} \)
29 \( 1 + (0.900 - 0.433i)T^{2} \)
31 \( 1 + (0.900 - 0.433i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.277 + 1.21i)T + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.623 + 0.781i)T^{2} \)
53 \( 1 + (0.222 + 0.974i)T^{2} \)
59 \( 1 + (0.277 - 0.347i)T + (-0.222 - 0.974i)T^{2} \)
61 \( 1 + (0.900 + 0.433i)T^{2} \)
67 \( 1 + (1.80 - 0.867i)T + (0.623 - 0.781i)T^{2} \)
71 \( 1 + (-0.623 + 0.781i)T^{2} \)
73 \( 1 + (-1.24 - 1.56i)T + (-0.222 + 0.974i)T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + (-0.0990 + 0.433i)T + (-0.900 - 0.433i)T^{2} \)
89 \( 1 + (0.277 - 1.21i)T + (-0.900 - 0.433i)T^{2} \)
97 \( 1 + (-0.400 - 0.193i)T + (0.623 + 0.781i)T^{2} \)
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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

−11.65082283909835953613100980492, −10.66983081564140114236225033809, −9.250278760956056078403472150322, −8.647777735199391449299104219627, −7.62267640569912267563634521260, −6.85615226917563959140846062622, −5.50650860779643453883056907470, −3.75231259594001000710590459664, −2.46539234798720148746249944003, −1.37756860400504313567997121253, 3.18072060962583270581830501526, 4.37661284175762954716656025174, 5.08123150497594287591359252110, 6.20001174570027212857977307040, 7.58783057384733848836844589784, 8.594720371501895251012502084908, 9.422360878353961131609822558751, 9.844860578269865541437697618358, 10.89139623536410374364275307326, 11.87367685940439359694721790159

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