# 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 $0$

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## 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}$$
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$$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

−10.52670437623373208646087056737, −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