Properties

Label 2-338-1.1-c5-0-29
Degree $2$
Conductor $338$
Sign $1$
Analytic cond. $54.2097$
Root an. cond. $7.36272$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·2-s + 16·4-s + 14·5-s + 170·7-s + 64·8-s − 243·9-s + 56·10-s + 250·11-s + 680·14-s + 256·16-s + 1.06e3·17-s − 972·18-s + 78·19-s + 224·20-s + 1.00e3·22-s + 1.57e3·23-s − 2.92e3·25-s + 2.72e3·28-s + 2.57e3·29-s + 8.65e3·31-s + 1.02e3·32-s + 4.24e3·34-s + 2.38e3·35-s − 3.88e3·36-s − 1.09e4·37-s + 312·38-s + 896·40-s + ⋯
L(s)  = 1  + 0.707·2-s + 1/2·4-s + 0.250·5-s + 1.31·7-s + 0.353·8-s − 9-s + 0.177·10-s + 0.622·11-s + 0.927·14-s + 1/4·16-s + 0.891·17-s − 0.707·18-s + 0.0495·19-s + 0.125·20-s + 0.440·22-s + 0.621·23-s − 0.937·25-s + 0.655·28-s + 0.569·29-s + 1.61·31-s + 0.176·32-s + 0.630·34-s + 0.328·35-s − 1/2·36-s − 1.31·37-s + 0.0350·38-s + 0.0885·40-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(54.2097\)
Root analytic conductor: \(7.36272\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :5/2),\ 1)\)

Particular Values

\(L(3)\) \(\approx\) \(4.210355428\)
\(L(\frac12)\) \(\approx\) \(4.210355428\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - p^{2} T \)
13 \( 1 \)
good3 \( 1 + p^{5} T^{2} \)
5 \( 1 - 14 T + p^{5} T^{2} \)
7 \( 1 - 170 T + p^{5} T^{2} \)
11 \( 1 - 250 T + p^{5} T^{2} \)
17 \( 1 - 1062 T + p^{5} T^{2} \)
19 \( 1 - 78 T + p^{5} T^{2} \)
23 \( 1 - 1576 T + p^{5} T^{2} \)
29 \( 1 - 2578 T + p^{5} T^{2} \)
31 \( 1 - 8654 T + p^{5} T^{2} \)
37 \( 1 + 10986 T + p^{5} T^{2} \)
41 \( 1 + 1050 T + p^{5} T^{2} \)
43 \( 1 + 5900 T + p^{5} T^{2} \)
47 \( 1 - 5962 T + p^{5} T^{2} \)
53 \( 1 - 29046 T + p^{5} T^{2} \)
59 \( 1 - 13922 T + p^{5} T^{2} \)
61 \( 1 + 32882 T + p^{5} T^{2} \)
67 \( 1 - 69566 T + p^{5} T^{2} \)
71 \( 1 - 50542 T + p^{5} T^{2} \)
73 \( 1 - 46750 T + p^{5} T^{2} \)
79 \( 1 + 19348 T + p^{5} T^{2} \)
83 \( 1 - 87438 T + p^{5} T^{2} \)
89 \( 1 + 94170 T + p^{5} T^{2} \)
97 \( 1 + 182786 T + p^{5} T^{2} \)
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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

−10.99424396948125200608536522354, −9.935486425855896917676733113341, −8.633932547382844396979031947778, −7.930025617070714970980513756793, −6.68861986266315716256592036399, −5.60166379418515331758729898248, −4.87236082953318653766574383275, −3.61172260155817529008055785806, −2.33522380792604885359597305251, −1.09570131594753959322816398447, 1.09570131594753959322816398447, 2.33522380792604885359597305251, 3.61172260155817529008055785806, 4.87236082953318653766574383275, 5.60166379418515331758729898248, 6.68861986266315716256592036399, 7.930025617070714970980513756793, 8.633932547382844396979031947778, 9.935486425855896917676733113341, 10.99424396948125200608536522354

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