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Properties

Label 2-336-112.109-c1-0-4
Degree $2$
Conductor $336$
Sign $0.845 - 0.534i$
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  + (0.0367 − 1.41i)2-s + (−0.965 + 0.258i)3-s + (−1.99 − 0.103i)4-s + (−0.377 − 0.101i)5-s + (0.330 + 1.37i)6-s + (−1.14 + 2.38i)7-s + (−0.220 + 2.81i)8-s + (0.866 − 0.499i)9-s + (−0.156 + 0.529i)10-s + (0.179 + 0.669i)11-s + (1.95 − 0.416i)12-s + (3.15 + 3.15i)13-s + (3.33 + 1.70i)14-s + 0.390·15-s + (3.97 + 0.415i)16-s + (0.143 − 0.249i)17-s + ⋯
L(s)  = 1  + (0.0259 − 0.999i)2-s + (−0.557 + 0.149i)3-s + (−0.998 − 0.0519i)4-s + (−0.168 − 0.0451i)5-s + (0.134 + 0.561i)6-s + (−0.432 + 0.901i)7-s + (−0.0779 + 0.996i)8-s + (0.288 − 0.166i)9-s + (−0.0495 + 0.167i)10-s + (0.0541 + 0.201i)11-s + (0.564 − 0.120i)12-s + (0.875 + 0.875i)13-s + (0.890 + 0.455i)14-s + 0.100·15-s + (0.994 + 0.103i)16-s + (0.0349 − 0.0604i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.845 - 0.534i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.728782 + 0.210917i\)
\(L(\frac12)\) \(\approx\) \(0.728782 + 0.210917i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.0367 + 1.41i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
7 \( 1 + (1.14 - 2.38i)T \)
good5 \( 1 + (0.377 + 0.101i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-0.179 - 0.669i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (-3.15 - 3.15i)T + 13iT^{2} \)
17 \( 1 + (-0.143 + 0.249i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.57 - 5.87i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.38 + 0.800i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.82 - 5.82i)T + 29iT^{2} \)
31 \( 1 + (2.04 - 3.53i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.27 + 0.340i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 0.356iT - 41T^{2} \)
43 \( 1 + (4.93 - 4.93i)T - 43iT^{2} \)
47 \( 1 + (-0.606 - 1.05i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.49 + 9.29i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (-3.14 - 11.7i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.822 - 3.06i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (9.46 - 2.53i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + 13.5iT - 71T^{2} \)
73 \( 1 + (-8.28 - 4.78i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-0.354 - 0.613i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.0770 - 0.0770i)T + 83iT^{2} \)
89 \( 1 + (-3.73 + 2.15i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−11.80404414334961522687570300846, −10.77479995350124318925982534380, −9.974230794071662153417831637574, −9.026597927599277075950142447285, −8.255933448900230970678103920498, −6.55438959803408293838430867809, −5.59522598990164776906820520296, −4.42282100466168419109749302005, −3.31290083889355177239114088789, −1.70823232898452017170723549052, 0.60124548843706476191872293315, 3.49609249879411182556977581914, 4.59538250591642480376880125526, 5.79689448479997397455825260000, 6.62201860807702461413863339550, 7.49744363160529290225846098896, 8.405056904503590808311080761805, 9.567352038149961550329766812828, 10.49611266127312891901872218759, 11.37724557555991414728646382717

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