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Properties

Label 2-338-13.12-c3-0-18
Degree $2$
Conductor $338$
Sign $0.722 + 0.691i$
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  − 2i·2-s + 0.405·3-s − 4·4-s + 6.36i·5-s − 0.811i·6-s + 2.55i·7-s + 8i·8-s − 26.8·9-s + 12.7·10-s − 26.1i·11-s − 1.62·12-s + 5.10·14-s + 2.58i·15-s + 16·16-s + 93.7·17-s + 53.6i·18-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.0780·3-s − 0.5·4-s + 0.569i·5-s − 0.0552i·6-s + 0.137i·7-s + 0.353i·8-s − 0.993·9-s + 0.402·10-s − 0.715i·11-s − 0.0390·12-s + 0.0973·14-s + 0.0444i·15-s + 0.250·16-s + 1.33·17-s + 0.702i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.722 + 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(338\)    =    \(2 \cdot 13^{2}\)
Sign: $0.722 + 0.691i$
Analytic conductor: \(19.9426\)
Root analytic conductor: \(4.46571\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{338} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 338,\ (\ :3/2),\ 0.722 + 0.691i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.698370825\)
\(L(\frac12)\) \(\approx\) \(1.698370825\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 2iT \)
13 \( 1 \)
good3 \( 1 - 0.405T + 27T^{2} \)
5 \( 1 - 6.36iT - 125T^{2} \)
7 \( 1 - 2.55iT - 343T^{2} \)
11 \( 1 + 26.1iT - 1.33e3T^{2} \)
17 \( 1 - 93.7T + 4.91e3T^{2} \)
19 \( 1 + 37.2iT - 6.85e3T^{2} \)
23 \( 1 - 104.T + 1.21e4T^{2} \)
29 \( 1 - 249.T + 2.43e4T^{2} \)
31 \( 1 - 278. iT - 2.97e4T^{2} \)
37 \( 1 - 10.9iT - 5.06e4T^{2} \)
41 \( 1 + 371. iT - 6.89e4T^{2} \)
43 \( 1 - 413.T + 7.95e4T^{2} \)
47 \( 1 + 238. iT - 1.03e5T^{2} \)
53 \( 1 + 424.T + 1.48e5T^{2} \)
59 \( 1 - 774. iT - 2.05e5T^{2} \)
61 \( 1 + 123.T + 2.26e5T^{2} \)
67 \( 1 + 881. iT - 3.00e5T^{2} \)
71 \( 1 - 118. iT - 3.57e5T^{2} \)
73 \( 1 + 209. iT - 3.89e5T^{2} \)
79 \( 1 + 532.T + 4.93e5T^{2} \)
83 \( 1 + 376. iT - 5.71e5T^{2} \)
89 \( 1 + 42.6iT - 7.04e5T^{2} \)
97 \( 1 - 639. iT - 9.12e5T^{2} \)
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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

−10.86603557112708849793568247720, −10.42778304327317542383022908900, −9.070268192690825095964093058144, −8.486487065459823048382529734529, −7.22281961381026513983673183433, −5.97375981480218879073122648243, −4.98648871879484606777859061549, −3.33872817404783937145454828001, −2.74284704546572871565166159698, −0.873667206228147517130389355106, 0.937018367015572268671032235307, 2.91693750457705846337818534357, 4.39954187606123304998138240213, 5.36771328300417166017375656125, 6.31205144556500458354625524485, 7.56011531358369396022832223982, 8.279651033657509543820840918535, 9.236175173491493525521584389571, 10.05682316083340078986685238240, 11.28091581496642658292213094913

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