Properties

Label 2-338-13.12-c7-0-38
Degree $2$
Conductor $338$
Sign $0.554 - 0.832i$
Analytic cond. $105.586$
Root an. cond. $10.2755$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 8i·2-s − 39·3-s − 64·4-s + 385i·5-s + 312i·6-s + 293i·7-s + 512i·8-s − 666·9-s + 3.08e3·10-s + 5.40e3i·11-s + 2.49e3·12-s + 2.34e3·14-s − 1.50e4i·15-s + 4.09e3·16-s + 2.10e4·17-s + 5.32e3i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.833·3-s − 0.5·4-s + 1.37i·5-s + 0.589i·6-s + 0.322i·7-s + 0.353i·8-s − 0.304·9-s + 0.973·10-s + 1.22i·11-s + 0.416·12-s + 0.228·14-s − 1.14i·15-s + 0.250·16-s + 1.03·17-s + 0.215i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(105.586\)
Root analytic conductor: \(10.2755\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :7/2),\ 0.554 - 0.832i)\)

Particular Values

\(L(4)\) \(\approx\) \(1.437175195\)
\(L(\frac12)\) \(\approx\) \(1.437175195\)
\(L(\frac{9}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 8iT \)
13 \( 1 \)
good3 \( 1 + 39T + 2.18e3T^{2} \)
5 \( 1 - 385iT - 7.81e4T^{2} \)
7 \( 1 - 293iT - 8.23e5T^{2} \)
11 \( 1 - 5.40e3iT - 1.94e7T^{2} \)
17 \( 1 - 2.10e4T + 4.10e8T^{2} \)
19 \( 1 + 2.73e4iT - 8.93e8T^{2} \)
23 \( 1 - 6.30e4T + 3.40e9T^{2} \)
29 \( 1 - 1.22e5T + 1.72e10T^{2} \)
31 \( 1 + 2.08e5iT - 2.75e10T^{2} \)
37 \( 1 - 4.42e5iT - 9.49e10T^{2} \)
41 \( 1 - 5.80e4iT - 1.94e11T^{2} \)
43 \( 1 - 2.02e5T + 2.71e11T^{2} \)
47 \( 1 + 5.88e5iT - 5.06e11T^{2} \)
53 \( 1 - 1.68e6T + 1.17e12T^{2} \)
59 \( 1 - 4.42e5iT - 2.48e12T^{2} \)
61 \( 1 + 1.08e6T + 3.14e12T^{2} \)
67 \( 1 - 3.44e6iT - 6.06e12T^{2} \)
71 \( 1 - 2.08e6iT - 9.09e12T^{2} \)
73 \( 1 + 5.93e6iT - 1.10e13T^{2} \)
79 \( 1 + 6.60e6T + 1.92e13T^{2} \)
83 \( 1 + 1.42e5iT - 2.71e13T^{2} \)
89 \( 1 - 6.98e6iT - 4.42e13T^{2} \)
97 \( 1 + 2.00e5iT - 8.07e13T^{2} \)
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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

−10.48747858526193700949457247822, −10.03094107538308766838197331089, −8.835422623981705546196043442798, −7.44712271987864896372210228205, −6.63456018076680221620036822272, −5.57056685770835917285746988240, −4.55108695281147407131610871493, −3.10153718910863019290318908148, −2.38652073080089570093399156078, −0.833569309101614455452504570622, 0.56367604085787851242333206516, 1.08181423423526317487075699708, 3.33068289906796800762471707999, 4.62412521713152335115867145626, 5.50139508253088040762192221584, 5.98634324063312431265013593240, 7.32796666816401280949529249477, 8.467802719725557571343247557711, 8.890014344999301908129098241912, 10.20109165540770234411833654823

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