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Properties

Label 2-338-13.3-c1-0-4
Degree $2$
Conductor $338$
Sign $0.0128 - 0.999i$
Analytic cond. $2.69894$
Root an. cond. $1.64284$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + 3·5-s + (−0.499 + 0.866i)6-s + (−1.5 + 2.59i)7-s − 0.999·8-s + (1 − 1.73i)9-s + (1.5 + 2.59i)10-s − 0.999·12-s − 3·14-s + (1.5 + 2.59i)15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + 2·18-s + (−3 + 5.19i)19-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + 1.34·5-s + (−0.204 + 0.353i)6-s + (−0.566 + 0.981i)7-s − 0.353·8-s + (0.333 − 0.577i)9-s + (0.474 + 0.821i)10-s − 0.288·12-s − 0.801·14-s + (0.387 + 0.670i)15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + 0.471·18-s + (−0.688 + 1.19i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.0128 - 0.999i$
Analytic conductor: \(2.69894\)
Root analytic conductor: \(1.64284\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (315, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.38457 + 1.36693i\)
\(L(\frac12)\) \(\approx\) \(1.38457 + 1.36693i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good3 \( 1 + (-0.5 - 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 - 3T + 5T^{2} \)
7 \( 1 + (1.5 - 2.59i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (1.5 + 2.59i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3T + 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (6 + 10.3i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.5 + 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 6T + 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-3 - 5.19i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−12.26395818682726632116059549512, −10.53990999773492967339176471733, −9.632752111841649690562806604446, −9.192205794854160367539989416599, −8.120769876406963313617146210040, −6.52630714919313276134151311773, −6.05073368177609842850232491163, −4.99783594429938193343624104777, −3.60231708309518999372236362043, −2.29470585204494744961248711066, 1.46212745385382789925937279808, 2.59365953319811390258767325425, 4.04817055877821881038495805772, 5.34852880142234575628543882209, 6.42917376145958731951745367094, 7.33979937657994471212318056198, 8.668468844428794053999956023319, 9.844311197465793993829672255421, 10.25064951354585636236305276574, 11.19697073523930542194592637034

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