Properties

Label 2-338-13.12-c5-0-44
Degree $2$
Conductor $338$
Sign $-0.554 + 0.832i$
Analytic cond. $54.2097$
Root an. cond. $7.36272$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 4i·2-s − 16·4-s − 14i·5-s + 170i·7-s + 64i·8-s − 243·9-s − 56·10-s + 250i·11-s + 680·14-s + 256·16-s − 1.06e3·17-s + 972i·18-s − 78i·19-s + 224i·20-s + 1.00e3·22-s − 1.57e3·23-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.5·4-s − 0.250i·5-s + 1.31i·7-s + 0.353i·8-s − 9-s − 0.177·10-s + 0.622i·11-s + 0.927·14-s + 0.250·16-s − 0.891·17-s + 0.707i·18-s − 0.0495i·19-s + 0.125i·20-s + 0.440·22-s − 0.621·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.554 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.554 + 0.832i$
Analytic conductor: \(54.2097\)
Root analytic conductor: \(7.36272\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :5/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(3)\) \(\approx\) \(0.9102551054\)
\(L(\frac12)\) \(\approx\) \(0.9102551054\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 4iT \)
13 \( 1 \)
good3 \( 1 + 243T^{2} \)
5 \( 1 + 14iT - 3.12e3T^{2} \)
7 \( 1 - 170iT - 1.68e4T^{2} \)
11 \( 1 - 250iT - 1.61e5T^{2} \)
17 \( 1 + 1.06e3T + 1.41e6T^{2} \)
19 \( 1 + 78iT - 2.47e6T^{2} \)
23 \( 1 + 1.57e3T + 6.43e6T^{2} \)
29 \( 1 - 2.57e3T + 2.05e7T^{2} \)
31 \( 1 + 8.65e3iT - 2.86e7T^{2} \)
37 \( 1 + 1.09e4iT - 6.93e7T^{2} \)
41 \( 1 - 1.05e3iT - 1.15e8T^{2} \)
43 \( 1 - 5.90e3T + 1.47e8T^{2} \)
47 \( 1 - 5.96e3iT - 2.29e8T^{2} \)
53 \( 1 - 2.90e4T + 4.18e8T^{2} \)
59 \( 1 - 1.39e4iT - 7.14e8T^{2} \)
61 \( 1 + 3.28e4T + 8.44e8T^{2} \)
67 \( 1 + 6.95e4iT - 1.35e9T^{2} \)
71 \( 1 + 5.05e4iT - 1.80e9T^{2} \)
73 \( 1 - 4.67e4iT - 2.07e9T^{2} \)
79 \( 1 + 1.93e4T + 3.07e9T^{2} \)
83 \( 1 + 8.74e4iT - 3.93e9T^{2} \)
89 \( 1 + 9.41e4iT - 5.58e9T^{2} \)
97 \( 1 - 1.82e5iT - 8.58e9T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.52670437623373208646087056737, −9.239953446829448406996669192116, −8.892308876267946561737915119487, −7.82292951064070360673471166116, −6.24841889776524464696973241268, −5.37594317746463049065230905162, −4.28608786042296956617955774636, −2.77788378281161062467015562218, −2.05398656724154479524106572920, −0.28443486533384484785915829457, 0.945742807088301908286213353092, 2.92839972830556281471333532105, 4.07336084094551142888530082638, 5.20502818647704413774253051217, 6.40385021274449852780132469812, 7.07004411714749347462172582586, 8.227509907046812445894721881830, 8.867212949967051823484845908910, 10.22547743065033254704907002742, 10.85042755733207413015030337649

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