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Properties

Label 2-336-84.11-c3-0-27
Degree $2$
Conductor $336$
Sign $0.642 - 0.766i$
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  + (3.03 + 4.21i)3-s + (5.91 − 3.41i)5-s + (13.7 + 12.4i)7-s + (−8.54 + 25.6i)9-s + (30.3 − 52.6i)11-s + 2.34·13-s + (32.3 + 14.5i)15-s + (72.5 + 41.8i)17-s + (14.3 − 8.28i)19-s + (−10.6 + 95.6i)21-s + (19.1 + 33.2i)23-s + (−39.1 + 67.8i)25-s + (−133. + 41.7i)27-s − 236. i·29-s + (−5.88 − 3.39i)31-s + ⋯
L(s)  = 1  + (0.584 + 0.811i)3-s + (0.528 − 0.305i)5-s + (0.741 + 0.670i)7-s + (−0.316 + 0.948i)9-s + (0.832 − 1.44i)11-s + 0.0500·13-s + (0.556 + 0.250i)15-s + (1.03 + 0.597i)17-s + (0.173 − 0.100i)19-s + (−0.110 + 0.993i)21-s + (0.173 + 0.301i)23-s + (−0.313 + 0.543i)25-s + (−0.954 + 0.297i)27-s − 1.51i·29-s + (−0.0341 − 0.0196i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.642 - 0.766i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (95, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.642 - 0.766i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.884320336\)
\(L(\frac12)\) \(\approx\) \(2.884320336\)
\(L(\frac{5}{2})\) 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 \( 1 + (-3.03 - 4.21i)T \)
7 \( 1 + (-13.7 - 12.4i)T \)
good5 \( 1 + (-5.91 + 3.41i)T + (62.5 - 108. i)T^{2} \)
11 \( 1 + (-30.3 + 52.6i)T + (-665.5 - 1.15e3i)T^{2} \)
13 \( 1 - 2.34T + 2.19e3T^{2} \)
17 \( 1 + (-72.5 - 41.8i)T + (2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-14.3 + 8.28i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-19.1 - 33.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + 236. iT - 2.43e4T^{2} \)
31 \( 1 + (5.88 + 3.39i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (-138. - 239. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 - 223. iT - 6.89e4T^{2} \)
43 \( 1 + 136. iT - 7.95e4T^{2} \)
47 \( 1 + (196. + 339. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-93.1 - 53.7i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (376. - 652. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-405. - 701. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (361. + 208. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 74.7T + 3.57e5T^{2} \)
73 \( 1 + (123. - 213. i)T + (-1.94e5 - 3.36e5i)T^{2} \)
79 \( 1 + (-570. + 329. i)T + (2.46e5 - 4.26e5i)T^{2} \)
83 \( 1 - 664.T + 5.71e5T^{2} \)
89 \( 1 + (-254. + 147. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + 1.30e3T + 9.12e5T^{2} \)
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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

−11.30021072964511051889563823904, −10.17166201503681905032061530516, −9.312042008548595083812379519460, −8.564845262127743465094401534583, −7.86284251237140192174354379645, −6.02238001863115950810706672042, −5.37325606046617154653783377213, −4.07643324226254151104442772368, −2.93431880478823615377060299195, −1.44302444183053392069147424798, 1.17010069726997686137671043923, 2.17992339816627498866175221058, 3.65922706963861134903936757098, 4.97664173174897721764647100327, 6.42702369541938247404448633134, 7.24014645981898475927454745060, 7.925624663520391132195898492422, 9.203296201152079884276593284171, 9.901330089133630849216593085143, 11.03166638220989704781937079646

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