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Properties

Label 2-338-13.10-c3-0-18
Degree $2$
Conductor $338$
Sign $-0.890 + 0.455i$
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  + (−1.73 + i)2-s + (−5.00 − 8.66i)3-s + (1.99 − 3.46i)4-s + 13.8i·5-s + (17.3 + 10.0i)6-s + (0.227 + 0.131i)7-s + 7.99i·8-s + (−36.5 + 63.3i)9-s + (−13.8 − 23.9i)10-s + (34.6 − 20.0i)11-s − 40.0·12-s − 0.524·14-s + (119. − 69.1i)15-s + (−8 − 13.8i)16-s + (39.8 − 69.0i)17-s − 146. i·18-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (−0.963 − 1.66i)3-s + (0.249 − 0.433i)4-s + 1.23i·5-s + (1.17 + 0.680i)6-s + (0.0122 + 0.00708i)7-s + 0.353i·8-s + (−1.35 + 2.34i)9-s + (−0.436 − 0.756i)10-s + (0.949 − 0.548i)11-s − 0.963·12-s − 0.0100·14-s + (2.06 − 1.18i)15-s + (−0.125 − 0.216i)16-s + (0.568 − 0.985i)17-s − 1.91i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4394913483\)
\(L(\frac12)\) \(\approx\) \(0.4394913483\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

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

−11.01635799674086847190296606771, −9.948538391168158322788713206624, −8.476733556266051172695346698280, −7.59926351366008627032753177833, −6.63237516965823256935895495246, −6.49021824984571591195566272536, −5.25360762134612938104966849889, −2.92057910463522026015217494556, −1.57810466337066613870949053854, −0.24563890090259189560100686482, 1.26260420774190604024092956249, 3.65162488406968872304946793175, 4.40883596566012560284651688499, 5.39820316317703974474113708464, 6.42992785668851320108156162628, 8.191233025805416066331721189072, 9.138428220151445807934707353790, 9.644287002681821397137759400225, 10.44545091719078221729258105401, 11.38935293743301248461499397370

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