Properties

Label 2-338-1.1-c9-0-50
Degree $2$
Conductor $338$
Sign $-1$
Analytic cond. $174.082$
Root an. cond. $13.1940$
Motivic weight $9$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 16·2-s − 156·3-s + 256·4-s − 870·5-s + 2.49e3·6-s + 952·7-s − 4.09e3·8-s + 4.65e3·9-s + 1.39e4·10-s + 5.61e4·11-s − 3.99e4·12-s − 1.52e4·14-s + 1.35e5·15-s + 6.55e4·16-s − 2.47e5·17-s − 7.44e4·18-s − 3.15e5·19-s − 2.22e5·20-s − 1.48e5·21-s − 8.98e5·22-s + 2.04e5·23-s + 6.38e5·24-s − 1.19e6·25-s + 2.34e6·27-s + 2.43e5·28-s − 3.84e6·29-s − 2.17e6·30-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.11·3-s + 1/2·4-s − 0.622·5-s + 0.786·6-s + 0.149·7-s − 0.353·8-s + 0.236·9-s + 0.440·10-s + 1.15·11-s − 0.555·12-s − 0.105·14-s + 0.692·15-s + 1/4·16-s − 0.719·17-s − 0.167·18-s − 0.555·19-s − 0.311·20-s − 0.166·21-s − 0.817·22-s + 0.152·23-s + 0.393·24-s − 0.612·25-s + 0.849·27-s + 0.0749·28-s − 1.00·29-s − 0.489·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(174.082\)
Root analytic conductor: \(13.1940\)
Motivic weight: \(9\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 338,\ (\ :9/2),\ -1)\)

Particular Values

\(L(5)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{11}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + p^{4} T \)
13 \( 1 \)
good3 \( 1 + 52 p T + p^{9} T^{2} \)
5 \( 1 + 174 p T + p^{9} T^{2} \)
7 \( 1 - 136 p T + p^{9} T^{2} \)
11 \( 1 - 56148 T + p^{9} T^{2} \)
17 \( 1 + 247662 T + p^{9} T^{2} \)
19 \( 1 + 315380 T + p^{9} T^{2} \)
23 \( 1 - 204504 T + p^{9} T^{2} \)
29 \( 1 + 3840450 T + p^{9} T^{2} \)
31 \( 1 - 1309408 T + p^{9} T^{2} \)
37 \( 1 + 4307078 T + p^{9} T^{2} \)
41 \( 1 + 1512042 T + p^{9} T^{2} \)
43 \( 1 - 33670604 T + p^{9} T^{2} \)
47 \( 1 - 10581072 T + p^{9} T^{2} \)
53 \( 1 - 16616214 T + p^{9} T^{2} \)
59 \( 1 + 112235100 T + p^{9} T^{2} \)
61 \( 1 + 33197218 T + p^{9} T^{2} \)
67 \( 1 - 121372252 T + p^{9} T^{2} \)
71 \( 1 - 387172728 T + p^{9} T^{2} \)
73 \( 1 + 255240074 T + p^{9} T^{2} \)
79 \( 1 - 492101840 T + p^{9} T^{2} \)
83 \( 1 - 457420236 T + p^{9} T^{2} \)
89 \( 1 - 31809510 T + p^{9} T^{2} \)
97 \( 1 - 673532062 T + p^{9} T^{2} \)
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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

−9.491465344170134967946180113164, −8.681672430833162749297321296427, −7.61788985340566614838224529898, −6.63496247203773092492264972052, −5.94093467628444970023801896674, −4.67970142295907655473049968751, −3.64061059252220698071583105026, −2.05012236499922882229758085185, −0.854314187281404098004802290684, 0, 0.854314187281404098004802290684, 2.05012236499922882229758085185, 3.64061059252220698071583105026, 4.67970142295907655473049968751, 5.94093467628444970023801896674, 6.63496247203773092492264972052, 7.61788985340566614838224529898, 8.681672430833162749297321296427, 9.491465344170134967946180113164

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