Properties

Label 2-338-13.12-c7-0-67
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8i·2-s − 87·3-s − 64·4-s + 321i·5-s − 696i·6-s + 181i·7-s − 512i·8-s + 5.38e3·9-s − 2.56e3·10-s − 7.78e3i·11-s + 5.56e3·12-s − 1.44e3·14-s − 2.79e4i·15-s + 4.09e3·16-s − 9.06e3·17-s + 4.30e4i·18-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.86·3-s − 0.5·4-s + 1.14i·5-s − 1.31i·6-s + 0.199i·7-s − 0.353i·8-s + 2.46·9-s − 0.812·10-s − 1.76i·11-s + 0.930·12-s − 0.141·14-s − 2.13i·15-s + 0.250·16-s − 0.447·17-s + 1.74i·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: \(1\)
Selberg data: \((2,\ 338,\ (\ :7/2),\ -0.554 + 0.832i)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 + 87T + 2.18e3T^{2} \)
5 \( 1 - 321iT - 7.81e4T^{2} \)
7 \( 1 - 181iT - 8.23e5T^{2} \)
11 \( 1 + 7.78e3iT - 1.94e7T^{2} \)
17 \( 1 + 9.06e3T + 4.10e8T^{2} \)
19 \( 1 + 3.71e4iT - 8.93e8T^{2} \)
23 \( 1 + 1.90e4T + 3.40e9T^{2} \)
29 \( 1 - 1.74e5T + 1.72e10T^{2} \)
31 \( 1 - 2.90e4iT - 2.75e10T^{2} \)
37 \( 1 + 3.23e5iT - 9.49e10T^{2} \)
41 \( 1 - 7.95e5iT - 1.94e11T^{2} \)
43 \( 1 - 3.14e5T + 2.71e11T^{2} \)
47 \( 1 - 4.47e5iT - 5.06e11T^{2} \)
53 \( 1 + 1.46e6T + 1.17e12T^{2} \)
59 \( 1 + 1.62e6iT - 2.48e12T^{2} \)
61 \( 1 + 2.39e6T + 3.14e12T^{2} \)
67 \( 1 + 6.40e4iT - 6.06e12T^{2} \)
71 \( 1 + 3.22e5iT - 9.09e12T^{2} \)
73 \( 1 - 4.45e6iT - 1.10e13T^{2} \)
79 \( 1 - 7.53e5T + 1.92e13T^{2} \)
83 \( 1 + 1.21e6iT - 2.71e13T^{2} \)
89 \( 1 + 3.39e6iT - 4.42e13T^{2} \)
97 \( 1 - 1.62e6iT - 8.07e13T^{2} \)
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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

−10.37935337488413392233323810722, −9.132640414681995362321294894913, −7.79332878846844165681004987439, −6.60686448897603331567363082071, −6.35468288978839135471083215696, −5.43546010140420970936348631065, −4.40582965660904932100389888056, −2.95943227866243295068341711485, −0.915897435920530964969315821238, 0, 1.05956049128530104561835225235, 1.85508830002195248241390960404, 4.19416401318390082760839990914, 4.68157249562387811649174062039, 5.52702163872793654536631412678, 6.64199861468449579946374845324, 7.74792235935024708327052934012, 9.161308939498770533435175978028, 10.14085417719354482936905972893

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