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Properties

Label 2-344-344.267-c0-0-0
Degree $2$
Conductor $344$
Sign $0.359 - 0.932i$
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  + (0.623 + 0.781i)2-s + (0.440 + 0.0663i)3-s + (−0.222 + 0.974i)4-s + (0.222 + 0.385i)6-s + (−0.900 + 0.433i)8-s + (−0.766 − 0.236i)9-s + (−0.162 − 0.712i)11-s + (−0.162 + 0.414i)12-s + (−0.900 − 0.433i)16-s + (1.57 − 1.07i)17-s + (−0.292 − 0.746i)18-s + (−1.40 + 0.432i)19-s + (0.455 − 0.571i)22-s + (−0.425 + 0.131i)24-s + (0.365 + 0.930i)25-s + ⋯
L(s)  = 1  + (0.623 + 0.781i)2-s + (0.440 + 0.0663i)3-s + (−0.222 + 0.974i)4-s + (0.222 + 0.385i)6-s + (−0.900 + 0.433i)8-s + (−0.766 − 0.236i)9-s + (−0.162 − 0.712i)11-s + (−0.162 + 0.414i)12-s + (−0.900 − 0.433i)16-s + (1.57 − 1.07i)17-s + (−0.292 − 0.746i)18-s + (−1.40 + 0.432i)19-s + (0.455 − 0.571i)22-s + (−0.425 + 0.131i)24-s + (0.365 + 0.930i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.359 - 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(344\)    =    \(2^{3} \cdot 43\)
Sign: $0.359 - 0.932i$
Analytic conductor: \(0.171678\)
Root analytic conductor: \(0.414340\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{344} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 344,\ (\ :0),\ 0.359 - 0.932i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.113437457\)
\(L(\frac12)\) \(\approx\) \(1.113437457\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 + (0.988 - 0.149i)T \)
good3 \( 1 + (-0.440 - 0.0663i)T + (0.955 + 0.294i)T^{2} \)
5 \( 1 + (-0.365 - 0.930i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.162 + 0.712i)T + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.988 + 0.149i)T^{2} \)
17 \( 1 + (-1.57 + 1.07i)T + (0.365 - 0.930i)T^{2} \)
19 \( 1 + (1.40 - 0.432i)T + (0.826 - 0.563i)T^{2} \)
23 \( 1 + (-0.0747 + 0.997i)T^{2} \)
29 \( 1 + (-0.955 + 0.294i)T^{2} \)
31 \( 1 + (0.733 + 0.680i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.0931 - 0.116i)T + (-0.222 + 0.974i)T^{2} \)
47 \( 1 + (0.900 + 0.433i)T^{2} \)
53 \( 1 + (0.988 - 0.149i)T^{2} \)
59 \( 1 + (-1.78 - 0.858i)T + (0.623 + 0.781i)T^{2} \)
61 \( 1 + (0.733 - 0.680i)T^{2} \)
67 \( 1 + (0.955 - 0.294i)T + (0.826 - 0.563i)T^{2} \)
71 \( 1 + (-0.0747 - 0.997i)T^{2} \)
73 \( 1 + (0.0747 - 0.997i)T + (-0.988 - 0.149i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1.23 + 0.185i)T + (0.955 + 0.294i)T^{2} \)
89 \( 1 + (0.147 + 0.0222i)T + (0.955 + 0.294i)T^{2} \)
97 \( 1 + (-0.440 - 1.92i)T + (-0.900 + 0.433i)T^{2} \)
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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

−12.01595673345253816592350359266, −11.27475700814544424357355940405, −9.885687648120551106255052058074, −8.769916798582093612227903401494, −8.177823093100918251333675311304, −7.13788737449842064950153862263, −5.97566826813073745432833589537, −5.19685768941239747670185455400, −3.72577290809194607888574344161, −2.83840507886494907723339930088, 1.98239415132721060239527734989, 3.15147958605964482868145086568, 4.34786639643909162847398346176, 5.49866943416819908845972230635, 6.51138637650700631125196564845, 7.993447560475870412800194706967, 8.847217039736433303122024831565, 10.01227667981747162599485762943, 10.62833133367456879443940982328, 11.67195388905537081679632605948

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