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Properties

Label 2-336-336.101-c1-0-34
Degree $2$
Conductor $336$
Sign $-0.389 + 0.920i$
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  + (−1.14 − 0.829i)2-s + (−0.575 + 1.63i)3-s + (0.622 + 1.90i)4-s + (0.667 − 2.48i)5-s + (2.01 − 1.39i)6-s + (−2.63 + 0.250i)7-s + (0.864 − 2.69i)8-s + (−2.33 − 1.88i)9-s + (−2.83 + 2.29i)10-s + (0.245 + 0.915i)11-s + (−3.46 − 0.0774i)12-s + (−0.958 − 0.958i)13-s + (3.22 + 1.89i)14-s + (3.68 + 2.52i)15-s + (−3.22 + 2.36i)16-s + (2.81 − 4.88i)17-s + ⋯
L(s)  = 1  + (−0.809 − 0.586i)2-s + (−0.332 + 0.943i)3-s + (0.311 + 0.950i)4-s + (0.298 − 1.11i)5-s + (0.822 − 0.568i)6-s + (−0.995 + 0.0946i)7-s + (0.305 − 0.952i)8-s + (−0.778 − 0.627i)9-s + (−0.894 + 0.726i)10-s + (0.0739 + 0.276i)11-s + (−0.999 − 0.0223i)12-s + (−0.265 − 0.265i)13-s + (0.861 + 0.507i)14-s + (0.950 + 0.651i)15-s + (−0.806 + 0.591i)16-s + (0.683 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.389 + 0.920i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.389 + 0.920i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (101, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.389 + 0.920i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.278135 - 0.419759i\)
\(L(\frac12)\) \(\approx\) \(0.278135 - 0.419759i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.14 + 0.829i)T \)
3 \( 1 + (0.575 - 1.63i)T \)
7 \( 1 + (2.63 - 0.250i)T \)
good5 \( 1 + (-0.667 + 2.48i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.245 - 0.915i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (0.958 + 0.958i)T + 13iT^{2} \)
17 \( 1 + (-2.81 + 4.88i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.531 + 1.98i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.15 + 5.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (6.53 + 6.53i)T + 29iT^{2} \)
31 \( 1 + (0.190 + 0.110i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.75 + 6.55i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 9.39iT - 41T^{2} \)
43 \( 1 + (0.165 + 0.165i)T + 43iT^{2} \)
47 \( 1 + (1.41 + 2.45i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.661 + 2.46i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.43 - 1.45i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.851 - 3.17i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.89 + 14.5i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 3.97T + 71T^{2} \)
73 \( 1 + (4.59 - 7.95i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-8.20 - 14.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (0.483 - 0.483i)T - 83iT^{2} \)
89 \( 1 + (-2.49 + 1.44i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 - 14.0iT - 97T^{2} \)
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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

−11.15466102808905827285315423290, −10.00163062911621899614434625536, −9.542581415136735537886665482782, −8.952120045254534084928375628452, −7.76134323514628275726489510785, −6.36373158439317200573913286201, −5.13159334205606077607625058645, −3.99770309278523629529649426277, −2.68666224088309782858916658601, −0.46753912048539838906950494128, 1.74644638489785056599766409752, 3.25697548097854257738012838182, 5.66351613289287221530699019584, 6.22640613848702817876350494132, 7.08662853758004527149753018918, 7.76195116308865236200078046910, 8.991681277526262116464378956408, 10.08366289984627961443587590138, 10.68760062743086219820403056976, 11.67899485499383198998058256663

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