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Properties

Label 2-336-12.11-c3-0-9
Degree $2$
Conductor $336$
Sign $-0.328 - 0.944i$
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  + (4.90 − 1.70i)3-s + 20.5i·5-s + 7i·7-s + (21.1 − 16.7i)9-s − 63.9·11-s + 56.5·13-s + (35.1 + 100. i)15-s + 81.6i·17-s + 22.4i·19-s + (11.9 + 34.3i)21-s − 114.·23-s − 298.·25-s + (75.2 − 118. i)27-s + 62.7i·29-s − 88.6i·31-s + ⋯
L(s)  = 1  + (0.944 − 0.328i)3-s + 1.84i·5-s + 0.377i·7-s + (0.783 − 0.620i)9-s − 1.75·11-s + 1.20·13-s + (0.605 + 1.73i)15-s + 1.16i·17-s + 0.270i·19-s + (0.124 + 0.356i)21-s − 1.03·23-s − 2.38·25-s + (0.536 − 0.844i)27-s + 0.401i·29-s − 0.513i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.328 - 0.944i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.328 - 0.944i$
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: \((2,\ 336,\ (\ :3/2),\ -0.328 - 0.944i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.059018158\)
\(L(\frac12)\) \(\approx\) \(2.059018158\)
\(L(\frac{5}{2})\) 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 \( 1 + (-4.90 + 1.70i)T \)
7 \( 1 - 7iT \)
good5 \( 1 - 20.5iT - 125T^{2} \)
11 \( 1 + 63.9T + 1.33e3T^{2} \)
13 \( 1 - 56.5T + 2.19e3T^{2} \)
17 \( 1 - 81.6iT - 4.91e3T^{2} \)
19 \( 1 - 22.4iT - 6.85e3T^{2} \)
23 \( 1 + 114.T + 1.21e4T^{2} \)
29 \( 1 - 62.7iT - 2.43e4T^{2} \)
31 \( 1 + 88.6iT - 2.97e4T^{2} \)
37 \( 1 + 51.3T + 5.06e4T^{2} \)
41 \( 1 - 165. iT - 6.89e4T^{2} \)
43 \( 1 - 393. iT - 7.95e4T^{2} \)
47 \( 1 - 164.T + 1.03e5T^{2} \)
53 \( 1 - 231. iT - 1.48e5T^{2} \)
59 \( 1 + 13.4T + 2.05e5T^{2} \)
61 \( 1 - 665.T + 2.26e5T^{2} \)
67 \( 1 + 837. iT - 3.00e5T^{2} \)
71 \( 1 - 175.T + 3.57e5T^{2} \)
73 \( 1 - 301.T + 3.89e5T^{2} \)
79 \( 1 - 1.04e3iT - 4.93e5T^{2} \)
83 \( 1 - 1.26e3T + 5.71e5T^{2} \)
89 \( 1 - 780. iT - 7.04e5T^{2} \)
97 \( 1 + 780.T + 9.12e5T^{2} \)
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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

−11.10830587815836721068207987134, −10.51432363271099996491414579786, −9.708011552145468082388829560623, −8.203039704472934745064211451509, −7.84382849946889780766600894700, −6.62904094813203458502130489267, −5.86257241873315520567606605458, −3.84317479739909584131120145222, −2.94875801268438841338068734913, −2.03596505910109140477091251367, 0.62652500995617171403182603631, 2.15077785817859108566317828305, 3.71418400810268272415945537964, 4.77296948848676881808063822252, 5.52336438532801622451624877524, 7.41088690433566779876658261106, 8.272345023491456736252582622082, 8.788843780180822478561403241712, 9.773183195194740684333182299181, 10.60454863379908750894806815036

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