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Properties

Label 2-7e3-7.5-c0-0-1
Degree $2$
Conductor $343$
Sign $1$
Analytic cond. $0.171179$
Root an. cond. $0.413738$
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 − 0.385i)2-s + (0.400 + 0.694i)4-s + 0.801·8-s + (−0.5 + 0.866i)9-s + (−0.623 − 1.07i)11-s + (−0.222 + 0.385i)16-s + (0.222 + 0.385i)18-s − 0.554·22-s + (0.900 − 1.56i)23-s + (−0.5 − 0.866i)25-s − 1.80·29-s + (0.5 + 0.866i)32-s − 0.801·36-s + (−0.623 + 1.07i)37-s − 0.445·43-s + (0.5 − 0.866i)44-s + ⋯
L(s)  = 1  + (0.222 − 0.385i)2-s + (0.400 + 0.694i)4-s + 0.801·8-s + (−0.5 + 0.866i)9-s + (−0.623 − 1.07i)11-s + (−0.222 + 0.385i)16-s + (0.222 + 0.385i)18-s − 0.554·22-s + (0.900 − 1.56i)23-s + (−0.5 − 0.866i)25-s − 1.80·29-s + (0.5 + 0.866i)32-s − 0.801·36-s + (−0.623 + 1.07i)37-s − 0.445·43-s + (0.5 − 0.866i)44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(343\)    =    \(7^{3}\)
Sign: $1$
Analytic conductor: \(0.171179\)
Root analytic conductor: \(0.413738\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{343} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 343,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9411867898\)
\(L(\frac12)\) \(\approx\) \(0.9411867898\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
good2 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
3 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.623 + 1.07i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 + 0.866i)T^{2} \)
23 \( 1 + (-0.900 + 1.56i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + 1.80T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.623 - 1.07i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 0.445T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.222 - 0.385i)T + (-0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 + 0.866i)T^{2} \)
67 \( 1 + (-0.900 - 1.56i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + 0.445T + T^{2} \)
73 \( 1 + (0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.222 + 0.385i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.59191969647579721494102111136, −10.98975530957011158610819559746, −10.26865179336989470622428930680, −8.658098058300965036579287932940, −8.121825180414732121805646038562, −7.07452138186878694058648731945, −5.81834351031608906882421092064, −4.64579142667226724826640170439, −3.28654117561486097271160266144, −2.30709613013944813478349193405, 1.88557120157370809454424333276, 3.57963858302676031684651185298, 5.10832186684189240103502552380, 5.78624312852348378221258953801, 7.01405828297814805034803974654, 7.61095019899390385848317200914, 9.198077887199528471603977606514, 9.778169318918153780057953620228, 10.93201719757518244117680253395, 11.60104865452746296596829284761

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