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Properties

Label 2-336-112.19-c1-0-13
Degree $2$
Conductor $336$
Sign $0.0111 + 0.999i$
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  + (−1.41 + 0.0322i)2-s + (−0.258 + 0.965i)3-s + (1.99 − 0.0911i)4-s + (−3.59 + 0.963i)5-s + (0.334 − 1.37i)6-s + (−1.51 + 2.17i)7-s + (−2.82 + 0.193i)8-s + (−0.866 − 0.499i)9-s + (5.05 − 1.47i)10-s + (1.57 − 5.88i)11-s + (−0.429 + 1.95i)12-s + (1.44 − 1.44i)13-s + (2.06 − 3.11i)14-s − 3.72i·15-s + (3.98 − 0.364i)16-s + (0.926 − 0.535i)17-s + ⋯
L(s)  = 1  + (−0.999 + 0.0227i)2-s + (−0.149 + 0.557i)3-s + (0.998 − 0.0455i)4-s + (−1.60 + 0.430i)5-s + (0.136 − 0.560i)6-s + (−0.571 + 0.820i)7-s + (−0.997 + 0.0683i)8-s + (−0.288 − 0.166i)9-s + (1.59 − 0.467i)10-s + (0.475 − 1.77i)11-s + (−0.123 + 0.563i)12-s + (0.400 − 0.400i)13-s + (0.552 − 0.833i)14-s − 0.961i·15-s + (0.995 − 0.0910i)16-s + (0.224 − 0.129i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0111 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0111 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.0111 + 0.999i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ 0.0111 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.175008 - 0.173072i\)
\(L(\frac12)\) \(\approx\) \(0.175008 - 0.173072i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.41 - 0.0322i)T \)
3 \( 1 + (0.258 - 0.965i)T \)
7 \( 1 + (1.51 - 2.17i)T \)
good5 \( 1 + (3.59 - 0.963i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.57 + 5.88i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-1.44 + 1.44i)T - 13iT^{2} \)
17 \( 1 + (-0.926 + 0.535i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.70 - 0.457i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-0.427 + 0.740i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.0115 + 0.0115i)T + 29iT^{2} \)
31 \( 1 + (3.35 + 5.81i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (1.15 + 4.32i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 3.89T + 41T^{2} \)
43 \( 1 + (9.03 + 9.03i)T + 43iT^{2} \)
47 \( 1 + (6.12 - 10.6i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-3.21 - 0.862i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (7.81 + 2.09i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (2.30 + 8.61i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (4.61 + 1.23i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 4.06T + 71T^{2} \)
73 \( 1 + (-7.61 - 13.1i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.61 - 0.932i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.97 + 4.97i)T + 83iT^{2} \)
89 \( 1 + (-0.259 + 0.449i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 1.22iT - 97T^{2} \)
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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

−11.25092018615743590853998740987, −10.57024847446963518678246254797, −9.279054526947300103283672279869, −8.541529575082185668298375761850, −7.85059782947173703277355170110, −6.56108924851700807643482945367, −5.70632113148532218105016305331, −3.72688701518445107726741414848, −3.03306530217546482159092797641, −0.25395448251277399283679138910, 1.44562771044322179921572771224, 3.45232488393580675214851547419, 4.60238279023233634969966199075, 6.61263897380887401262947408748, 7.17149219042097342763001974363, 7.900657094500560030510851714050, 8.863226810009829382716592506262, 9.903911678658834385451632710656, 10.87670777892813579823242332188, 11.83999713615092886163114738051

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