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Properties

Label 2-338-13.9-c3-0-12
Degree $2$
Conductor $338$
Sign $0.611 - 0.791i$
Analytic cond. $19.9426$
Root an. cond. $4.46571$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−1 + 1.73i)2-s + (−5.00 + 8.66i)3-s + (−1.99 − 3.46i)4-s − 13.8·5-s + (−10.0 − 17.3i)6-s + (0.131 + 0.227i)7-s + 7.99·8-s + (−36.5 − 63.3i)9-s + (13.8 − 23.9i)10-s + (−20.0 + 34.6i)11-s + 40.0·12-s − 0.524·14-s + (69.1 − 119. i)15-s + (−8 + 13.8i)16-s + (−39.8 − 69.0i)17-s + 146.·18-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.963 + 1.66i)3-s + (−0.249 − 0.433i)4-s − 1.23·5-s + (−0.680 − 1.17i)6-s + (0.00708 + 0.0122i)7-s + 0.353·8-s + (−1.35 − 2.34i)9-s + (0.436 − 0.756i)10-s + (−0.548 + 0.949i)11-s + 0.963·12-s − 0.0100·14-s + (1.18 − 2.06i)15-s + (−0.125 + 0.216i)16-s + (−0.568 − 0.985i)17-s + 1.91·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.611 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.611 - 0.791i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (191, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ 0.611 - 0.791i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2727391137\)
\(L(\frac12)\) \(\approx\) \(0.2727391137\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1 - 1.73i)T \)
13 \( 1 \)
good3 \( 1 + (5.00 - 8.66i)T + (-13.5 - 23.3i)T^{2} \)
5 \( 1 + 13.8T + 125T^{2} \)
7 \( 1 + (-0.131 - 0.227i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (20.0 - 34.6i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (39.8 + 69.0i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-11.2 - 19.5i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-32.7 + 56.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (20.0 - 34.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 113.T + 2.97e4T^{2} \)
37 \( 1 + (60.5 - 104. i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (198. - 343. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (-137. - 238. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 440.T + 1.03e5T^{2} \)
53 \( 1 + 615.T + 1.48e5T^{2} \)
59 \( 1 + (-115. - 199. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-54.7 - 94.8i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-110. + 191. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-161. - 279. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 323.T + 3.89e5T^{2} \)
79 \( 1 - 743.T + 4.93e5T^{2} \)
83 \( 1 - 539.T + 5.71e5T^{2} \)
89 \( 1 + (-632. + 1.09e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (163. + 282. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−11.20558828179706041019016456171, −10.22359272260450134354508409751, −9.546018803563917546602329984076, −8.564721314692087588023115697690, −7.41440033446755901545908698119, −6.34309796833316313935361459711, −4.95684878472353706415164311882, −4.61354576574586526874163579798, −3.37202537418776790414338727218, −0.23398107623857426181468877493, 0.61157026520456943401805004059, 2.00578520986497661156765611700, 3.51942958877205766270335195711, 5.13573600330330407670012301788, 6.28112027910182137160925939962, 7.37491140334800760700501252003, 7.965683845798451598853938680737, 8.770802337384516801531145246427, 10.70618873762043459410769710204, 11.12243141640079619532216087920

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