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Properties

Label 2-336-28.19-c1-0-3
Degree $2$
Conductor $336$
Sign $0.832 - 0.553i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)3-s + (3 + 1.73i)5-s + (−0.5 − 2.59i)7-s + (−0.499 + 0.866i)9-s + (3 − 1.73i)11-s + 1.73i·13-s + 3.46i·15-s + (−2.5 + 4.33i)19-s + (2 − 1.73i)21-s + (−6 − 3.46i)23-s + (3.5 + 6.06i)25-s − 0.999·27-s + (2.5 + 4.33i)31-s + (3 + 1.73i)33-s + (3 − 8.66i)35-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (1.34 + 0.774i)5-s + (−0.188 − 0.981i)7-s + (−0.166 + 0.288i)9-s + (0.904 − 0.522i)11-s + 0.480i·13-s + 0.894i·15-s + (−0.573 + 0.993i)19-s + (0.436 − 0.377i)21-s + (−1.25 − 0.722i)23-s + (0.700 + 1.21i)25-s − 0.192·27-s + (0.449 + 0.777i)31-s + (0.522 + 0.301i)33-s + (0.507 − 1.46i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.832 - 0.553i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.67800 + 0.507115i\)
\(L(\frac12)\) \(\approx\) \(1.67800 + 0.507115i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (-3 - 1.73i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 1.73iT - 13T^{2} \)
17 \( 1 + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.5 - 4.33i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (6 + 3.46i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.5 + 9.52i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 - 3.46iT - 41T^{2} \)
43 \( 1 - 8.66iT - 43T^{2} \)
47 \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (12 + 6.92i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (7.5 - 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 + (4.5 - 2.59i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (10.5 + 6.06i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 - 18T + 83T^{2} \)
89 \( 1 + (-6 - 3.46i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + 6.92iT - 97T^{2} \)
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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

−11.34867077618516947519672883509, −10.47641176758024532273549197545, −9.927871008621372283969602023539, −9.109433634705778217341884943107, −7.88442733305591878489796024493, −6.52643661710426782187699777452, −6.07341180484282196462216718502, −4.42272945846061880247644676667, −3.35989489619348189556796851941, −1.87462165767029376235891253597, 1.58199541757540501807887980487, 2.65738939984752821013330157489, 4.53932921393914051605084982138, 5.81035001421093663734180988720, 6.33809972791854056707354639362, 7.76890664811245694987480613188, 9.008654293752635906015960744076, 9.275575648154472178810909426504, 10.31505460159617154616572448919, 11.83959352537171871881558092675

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