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Properties

Label 2-336-12.11-c3-0-22
Degree $2$
Conductor $336$
Sign $0.0571 + 0.998i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
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  + (−2.33 + 4.64i)3-s + 12.0i·5-s − 7i·7-s + (−16.0 − 21.6i)9-s − 33.9·11-s − 13.8·13-s + (−55.7 − 28.0i)15-s + 28.7i·17-s − 75.7i·19-s + (32.4 + 16.3i)21-s − 104.·23-s − 19.1·25-s + (138. − 23.9i)27-s + 242. i·29-s − 316. i·31-s + ⋯
L(s)  = 1  + (−0.449 + 0.893i)3-s + 1.07i·5-s − 0.377i·7-s + (−0.595 − 0.803i)9-s − 0.931·11-s − 0.295·13-s + (−0.959 − 0.482i)15-s + 0.409i·17-s − 0.914i·19-s + (0.337 + 0.169i)21-s − 0.945·23-s − 0.153·25-s + (0.985 − 0.170i)27-s + 1.55i·29-s − 1.83i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0571 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0571 + 0.998i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.0571 + 0.998i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.2921501467\)
\(L(\frac12)\) \(\approx\) \(0.2921501467\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (2.33 - 4.64i)T \)
7 \( 1 + 7iT \)
good5 \( 1 - 12.0iT - 125T^{2} \)
11 \( 1 + 33.9T + 1.33e3T^{2} \)
13 \( 1 + 13.8T + 2.19e3T^{2} \)
17 \( 1 - 28.7iT - 4.91e3T^{2} \)
19 \( 1 + 75.7iT - 6.85e3T^{2} \)
23 \( 1 + 104.T + 1.21e4T^{2} \)
29 \( 1 - 242. iT - 2.43e4T^{2} \)
31 \( 1 + 316. iT - 2.97e4T^{2} \)
37 \( 1 - 303.T + 5.06e4T^{2} \)
41 \( 1 + 382. iT - 6.89e4T^{2} \)
43 \( 1 + 12.0iT - 7.95e4T^{2} \)
47 \( 1 + 314.T + 1.03e5T^{2} \)
53 \( 1 + 418. iT - 1.48e5T^{2} \)
59 \( 1 + 477.T + 2.05e5T^{2} \)
61 \( 1 - 112.T + 2.26e5T^{2} \)
67 \( 1 + 180. iT - 3.00e5T^{2} \)
71 \( 1 + 25.4T + 3.57e5T^{2} \)
73 \( 1 + 349.T + 3.89e5T^{2} \)
79 \( 1 + 452. iT - 4.93e5T^{2} \)
83 \( 1 - 1.08e3T + 5.71e5T^{2} \)
89 \( 1 - 816. iT - 7.04e5T^{2} \)
97 \( 1 + 1.07e3T + 9.12e5T^{2} \)
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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

−10.81288624767190385677603728734, −10.20224567387202618141858539546, −9.318666432446198113378869630332, −7.998793639369897496970404774848, −6.95053629612053930692898319317, −5.96012006134616912867919198125, −4.85371974459793485135710289124, −3.68893897531970737866438304860, −2.56754214292077892766823146903, −0.11329559199937655191448582440, 1.31253728958003964768620798315, 2.65395294545415479996169579510, 4.61146595244043337763503208858, 5.47639312538001539353421645103, 6.38045040129271996083346136350, 7.80858816171667067081055676318, 8.213012221054151929521636692088, 9.439207609750054593809670255032, 10.48164944885976883868545890525, 11.66671143300290369246818714296

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