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Properties

Label 2-336-21.11-c2-0-18
Degree $2$
Conductor $336$
Sign $0.587 + 0.809i$
Analytic cond. $9.15533$
Root an. cond. $3.02577$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.81 − 1.04i)3-s + (5.12 − 2.95i)5-s + 7·7-s + (6.81 + 5.88i)9-s + (−5.12 − 2.95i)11-s − 6·13-s + (−17.5 + 2.95i)15-s + (5.12 + 2.95i)17-s + (11.5 + 19.9i)19-s + (−19.6 − 7.32i)21-s + (35.8 − 20.7i)23-s + (4.99 − 8.66i)25-s + (−13 − 23.6i)27-s − 47.3i·29-s + (19.5 − 33.7i)31-s + ⋯
L(s)  = 1  + (−0.937 − 0.348i)3-s + (1.02 − 0.591i)5-s + 7-s + (0.756 + 0.653i)9-s + (−0.465 − 0.268i)11-s − 0.461·13-s + (−1.16 + 0.197i)15-s + (0.301 + 0.174i)17-s + (0.605 + 1.04i)19-s + (−0.937 − 0.348i)21-s + (1.55 − 0.900i)23-s + (0.199 − 0.346i)25-s + (−0.481 − 0.876i)27-s − 1.63i·29-s + (0.629 − 1.08i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.587 + 0.809i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.587 + 0.809i$
Analytic conductor: \(9.15533\)
Root analytic conductor: \(3.02577\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (305, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1),\ 0.587 + 0.809i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.42522 - 0.726767i\)
\(L(\frac12)\) \(\approx\) \(1.42522 - 0.726767i\)
\(L(2)\) 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 \( 1 + (2.81 + 1.04i)T \)
7 \( 1 - 7T \)
good5 \( 1 + (-5.12 + 2.95i)T + (12.5 - 21.6i)T^{2} \)
11 \( 1 + (5.12 + 2.95i)T + (60.5 + 104. i)T^{2} \)
13 \( 1 + 6T + 169T^{2} \)
17 \( 1 + (-5.12 - 2.95i)T + (144.5 + 250. i)T^{2} \)
19 \( 1 + (-11.5 - 19.9i)T + (-180.5 + 312. i)T^{2} \)
23 \( 1 + (-35.8 + 20.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + 47.3iT - 841T^{2} \)
31 \( 1 + (-19.5 + 33.7i)T + (-480.5 - 832. i)T^{2} \)
37 \( 1 + (23.5 + 40.7i)T + (-684.5 + 1.18e3i)T^{2} \)
41 \( 1 - 1.68e3T^{2} \)
43 \( 1 - 22T + 1.84e3T^{2} \)
47 \( 1 + (46.1 - 26.6i)T + (1.10e3 - 1.91e3i)T^{2} \)
53 \( 1 + (-46.1 - 26.6i)T + (1.40e3 + 2.43e3i)T^{2} \)
59 \( 1 + (87.0 + 50.2i)T + (1.74e3 + 3.01e3i)T^{2} \)
61 \( 1 + (-40.5 - 70.1i)T + (-1.86e3 + 3.22e3i)T^{2} \)
67 \( 1 + (-15.5 + 26.8i)T + (-2.24e3 - 3.88e3i)T^{2} \)
71 \( 1 - 94.6iT - 5.04e3T^{2} \)
73 \( 1 + (-8.5 + 14.7i)T + (-2.66e3 - 4.61e3i)T^{2} \)
79 \( 1 + (4.5 + 7.79i)T + (-3.12e3 + 5.40e3i)T^{2} \)
83 \( 1 + 47.3iT - 6.88e3T^{2} \)
89 \( 1 + (76.8 - 44.3i)T + (3.96e3 - 6.85e3i)T^{2} \)
97 \( 1 - 82T + 9.40e3T^{2} \)
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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

−11.22799094891585671110027010966, −10.35714869273755704129173371979, −9.512767610629731907277347949739, −8.243696968035302039260589854820, −7.38903821635729287911530657492, −6.00616644643301006222185572005, −5.39580350803358917174411727759, −4.48648506216732112071238294756, −2.22749338205446606199014036415, −0.986619909247010685214120289553, 1.40676459833500292294384032035, 3.02987239832140530192295336463, 5.03030738172243988974143934459, 5.18287401785605762553926936057, 6.66256115499451033084996184513, 7.34446780973243824290856334649, 8.903077899672403838237453096426, 9.860541276293302715088638470444, 10.61578373479664510692192896950, 11.28626366574210421588326380822

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