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Properties

Label 2-336-21.5-c1-0-12
Degree $2$
Conductor $336$
Sign $0.553 + 0.832i$
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  + (1.5 − 0.866i)3-s + (−0.5 − 2.59i)7-s + (1.5 − 2.59i)9-s − 1.73i·13-s + (4.5 + 2.59i)19-s + (−3 − 3.46i)21-s + (2.5 + 4.33i)25-s − 5.19i·27-s + (−7.5 + 4.33i)31-s + (−0.5 + 0.866i)37-s + (−1.49 − 2.59i)39-s + 5·43-s + (−6.5 + 2.59i)49-s + 9·57-s + (6 + 3.46i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.188 − 0.981i)7-s + (0.5 − 0.866i)9-s − 0.480i·13-s + (1.03 + 0.596i)19-s + (−0.654 − 0.755i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (−1.34 + 0.777i)31-s + (−0.0821 + 0.142i)37-s + (−0.240 − 0.416i)39-s + 0.762·43-s + (−0.928 + 0.371i)49-s + 1.19·57-s + (0.768 + 0.443i)61-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.553 + 0.832i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.553 + 0.832i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50911 - 0.808653i\)
\(L(\frac12)\) \(\approx\) \(1.50911 - 0.808653i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.5 + 0.866i)T \)
7 \( 1 + (0.5 + 2.59i)T \)
good5 \( 1 + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.73iT - 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-4.5 - 2.59i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + (7.5 - 4.33i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 5T + 43T^{2} \)
47 \( 1 + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (13.5 - 7.79i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.5 - 11.2i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−11.41941785603222522887957765342, −10.33981794205294891009798878631, −9.533009708702607669203660333621, −8.525240011691245440657276104083, −7.49347509077663509375304093656, −6.96992794580905081562518928567, −5.53399340368707844810123254047, −3.97616432998908503075031220488, −3.04444386945885656669642568983, −1.29473696278255747630958387854, 2.18030472597969985171433190390, 3.29200566794896011726735625410, 4.59530871416190169385855182024, 5.69065440049857888425647511780, 7.04881314349125167750816694374, 8.107002892703647091153826134274, 9.089326327690703632501222123323, 9.531179361467725547520808615724, 10.69686699709282516646201134262, 11.67660316037631171716828806208

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