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Properties

Label 2-336-112.109-c1-0-20
Degree $2$
Conductor $336$
Sign $0.301 - 0.953i$
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  + (0.608 + 1.27i)2-s + (0.965 − 0.258i)3-s + (−1.26 + 1.55i)4-s + (2.60 + 0.698i)5-s + (0.917 + 1.07i)6-s + (2.64 + 0.152i)7-s + (−2.74 − 0.665i)8-s + (0.866 − 0.499i)9-s + (0.692 + 3.75i)10-s + (−0.429 − 1.60i)11-s + (−0.815 + 1.82i)12-s + (−4.38 − 4.38i)13-s + (1.41 + 3.46i)14-s + 2.69·15-s + (−0.822 − 3.91i)16-s + (0.139 − 0.241i)17-s + ⋯
L(s)  = 1  + (0.429 + 0.902i)2-s + (0.557 − 0.149i)3-s + (−0.630 + 0.776i)4-s + (1.16 + 0.312i)5-s + (0.374 + 0.439i)6-s + (0.998 + 0.0577i)7-s + (−0.971 − 0.235i)8-s + (0.288 − 0.166i)9-s + (0.219 + 1.18i)10-s + (−0.129 − 0.483i)11-s + (−0.235 + 0.527i)12-s + (−1.21 − 1.21i)13-s + (0.377 + 0.926i)14-s + 0.696·15-s + (−0.205 − 0.978i)16-s + (0.0337 − 0.0585i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.301 - 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.301 - 0.953i$
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: \((2,\ 336,\ (\ :1/2),\ 0.301 - 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75579 + 1.28558i\)
\(L(\frac12)\) \(\approx\) \(1.75579 + 1.28558i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.608 - 1.27i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
7 \( 1 + (-2.64 - 0.152i)T \)
good5 \( 1 + (-2.60 - 0.698i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (0.429 + 1.60i)T + (-9.52 + 5.5i)T^{2} \)
13 \( 1 + (4.38 + 4.38i)T + 13iT^{2} \)
17 \( 1 + (-0.139 + 0.241i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.76 - 6.59i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (1.62 - 0.940i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.52 - 1.52i)T + 29iT^{2} \)
31 \( 1 + (4.10 - 7.10i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.25 + 0.604i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 9.20iT - 41T^{2} \)
43 \( 1 + (-2.19 + 2.19i)T - 43iT^{2} \)
47 \( 1 + (4.05 + 7.01i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.09 + 4.08i)T + (-45.8 + 26.5i)T^{2} \)
59 \( 1 + (2.97 + 11.1i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (0.490 - 1.82i)T + (-52.8 - 30.5i)T^{2} \)
67 \( 1 + (3.94 - 1.05i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.1iT - 71T^{2} \)
73 \( 1 + (-4.55 - 2.62i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.68 - 13.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.79 - 2.79i)T + 83iT^{2} \)
89 \( 1 + (-15.5 + 8.99i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 4.17T + 97T^{2} \)
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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

−12.15297633651370312958960250631, −10.56806399633655186399405026523, −9.805916376273015559317748490816, −8.615568736677198879933021448889, −7.934279568575430374565652632662, −6.98785465958225886281656837107, −5.71683388222429842167661289093, −5.14265685470741386046202289995, −3.53194443014965170356785026802, −2.16134345836955483522585482271, 1.79476480018025580265544719625, 2.50908194649013025626892208866, 4.46787169654470676518086541063, 4.89486862328165990442816627899, 6.28369126170837654692345242303, 7.69757938780172919925478047930, 9.159935762488886896188984815749, 9.403913097613623957035876808345, 10.43442765469772394236756410503, 11.39114806953811536211098717248

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