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Properties

Label 2-336-336.101-c1-0-0
Degree $2$
Conductor $336$
Sign $-0.811 - 0.583i$
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  + (−0.838 + 1.13i)2-s + (−1.71 − 0.268i)3-s + (−0.592 − 1.91i)4-s + (0.392 − 1.46i)5-s + (1.74 − 1.72i)6-s + (−2.54 − 0.730i)7-s + (2.67 + 0.928i)8-s + (2.85 + 0.918i)9-s + (1.33 + 1.67i)10-s + (1.37 + 5.12i)11-s + (0.501 + 3.42i)12-s + (−3.27 − 3.27i)13-s + (2.96 − 2.28i)14-s + (−1.06 + 2.40i)15-s + (−3.29 + 2.26i)16-s + (−1.26 + 2.19i)17-s + ⋯
L(s)  = 1  + (−0.593 + 0.805i)2-s + (−0.987 − 0.154i)3-s + (−0.296 − 0.955i)4-s + (0.175 − 0.655i)5-s + (0.710 − 0.703i)6-s + (−0.961 − 0.276i)7-s + (0.944 + 0.328i)8-s + (0.952 + 0.306i)9-s + (0.423 + 0.530i)10-s + (0.413 + 1.54i)11-s + (0.144 + 0.989i)12-s + (−0.907 − 0.907i)13-s + (0.792 − 0.609i)14-s + (−0.275 + 0.620i)15-s + (−0.824 + 0.565i)16-s + (−0.307 + 0.532i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.811 - 0.583i$
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: \((2,\ 336,\ (\ :1/2),\ -0.811 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0979689 + 0.304008i\)
\(L(\frac12)\) \(\approx\) \(0.0979689 + 0.304008i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.838 - 1.13i)T \)
3 \( 1 + (1.71 + 0.268i)T \)
7 \( 1 + (2.54 + 0.730i)T \)
good5 \( 1 + (-0.392 + 1.46i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.37 - 5.12i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (3.27 + 3.27i)T + 13iT^{2} \)
17 \( 1 + (1.26 - 2.19i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.71 - 6.39i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.22 - 3.86i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.58 + 4.58i)T + 29iT^{2} \)
31 \( 1 + (2.15 + 1.24i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.82 - 6.82i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 - 8.27iT - 41T^{2} \)
43 \( 1 + (4.61 + 4.61i)T + 43iT^{2} \)
47 \( 1 + (-2.02 - 3.50i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.05 - 7.65i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.77 - 0.743i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (1.31 - 4.90i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (1.50 + 5.62i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 0.414T + 71T^{2} \)
73 \( 1 + (-4.33 + 7.50i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.52 - 6.09i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.44 + 1.44i)T - 83iT^{2} \)
89 \( 1 + (-2.58 + 1.49i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 2.84iT - 97T^{2} \)
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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

−12.09949043570708453257264688880, −10.64026724115238471690306041916, −9.900948993838325999055631520421, −9.409772876984959056726151903620, −7.87609298115269027795975735258, −7.12100462000394082460674666384, −6.18800526962163202252209067249, −5.27823319131744566675835415603, −4.27211029312872516044566738264, −1.54960705793598649267844269796, 0.31251082600061154217434715403, 2.53495383582265891405933313391, 3.73608327442711162847144563079, 5.13366912548266589427997415107, 6.64921169499912631050515094284, 6.99974310452438330548655015378, 8.907976456745080334847873454403, 9.347843040802238857774828584666, 10.55135202742972992993183307742, 11.04604043786056906550360588777

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