↑

Properties

Label 2-351-117.103-c1-0-8
Degree $2$
Conductor $351$
Sign $0.881 + 0.472i$
Analytic cond. $2.80274$
Root an. cond. $1.67414$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (1.41 − 0.818i)2-s + (0.338 − 0.587i)4-s + (0.950 + 0.548i)5-s + (2.77 − 1.60i)7-s + 2.16i·8-s + 1.79·10-s + (−1.52 + 0.879i)11-s + (1.37 − 3.33i)13-s + (2.62 − 4.54i)14-s + (2.44 + 4.24i)16-s − 1.47·17-s + 3.61i·19-s + (0.644 − 0.371i)20-s + (−1.43 + 2.49i)22-s + (2.34 − 4.06i)23-s + ⋯
L(s)  = 1  + (1.00 − 0.578i)2-s + (0.169 − 0.293i)4-s + (0.425 + 0.245i)5-s + (1.05 − 0.606i)7-s + 0.764i·8-s + 0.567·10-s + (−0.459 + 0.265i)11-s + (0.381 − 0.924i)13-s + (0.701 − 1.21i)14-s + (0.612 + 1.06i)16-s − 0.357·17-s + 0.829i·19-s + (0.144 − 0.0831i)20-s + (−0.306 + 0.531i)22-s + (0.489 − 0.847i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 351 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(351\)    =    \(3^{3} \cdot 13\)
Sign: $0.881 + 0.472i$
Analytic conductor: \(2.80274\)
Root analytic conductor: \(1.67414\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{351} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 351,\ (\ :1/2),\ 0.881 + 0.472i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.32895 - 0.585052i\)
\(L(\frac12)\) \(\approx\) \(2.32895 - 0.585052i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-1.37 + 3.33i)T \)
good2 \( 1 + (-1.41 + 0.818i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + (-0.950 - 0.548i)T + (2.5 + 4.33i)T^{2} \)
7 \( 1 + (-2.77 + 1.60i)T + (3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.52 - 0.879i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 1.47T + 17T^{2} \)
19 \( 1 - 3.61iT - 19T^{2} \)
23 \( 1 + (-2.34 + 4.06i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.959 + 1.66i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (5.68 + 3.28i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 11.6iT - 37T^{2} \)
41 \( 1 + (4.68 + 2.70i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-0.889 - 1.54i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (8.90 - 5.13i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + 11.7T + 53T^{2} \)
59 \( 1 + (-4.78 - 2.76i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.985 + 1.70i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-7.15 - 4.13i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + 5.84iT - 71T^{2} \)
73 \( 1 - 1.24iT - 73T^{2} \)
79 \( 1 + (0.242 + 0.420i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-13.4 + 7.75i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 - 13.4iT - 89T^{2} \)
97 \( 1 + (-5.15 + 2.97i)T + (48.5 - 84.0i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.38834094195597969982928570736, −10.77654661858257048914639565769, −9.962056927494064701982604600361, −8.360804921925356040331136188631, −7.80544363261147771879614669034, −6.28638630407156682960495312848, −5.18450466171961512495008135947, −4.38026382964945131912428197739, −3.15143615238460692876103761911, −1.84760556867096891918722721021, 1.82915898783126019069910290989, 3.62965845432739858250991193976, 4.98249132345931705729535927056, 5.37304482828002552972717405414, 6.53197900073851362209588794375, 7.53724101565786640189733706194, 8.819400024502104567828654478349, 9.481036626165301909755290203154, 10.93732088676993273530392068914, 11.59233586519771921965868061267

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