Properties

Label 2-315-9.7-c1-0-10
Degree $2$
Conductor $315$
Sign $0.839 - 0.543i$
Analytic cond. $2.51528$
Root an. cond. $1.58596$
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  + (0.392 − 0.680i)2-s + (1.02 + 1.39i)3-s + (0.691 + 1.19i)4-s + (−0.5 − 0.866i)5-s + (1.35 − 0.153i)6-s + (−0.5 + 0.866i)7-s + 2.65·8-s + (−0.880 + 2.86i)9-s − 0.785·10-s + (1.09 − 1.89i)11-s + (−0.956 + 2.19i)12-s + (0.415 + 0.719i)13-s + (0.392 + 0.680i)14-s + (0.691 − 1.58i)15-s + (−0.340 + 0.589i)16-s + 0.994·17-s + ⋯
L(s)  = 1  + (0.277 − 0.480i)2-s + (0.594 + 0.804i)3-s + (0.345 + 0.599i)4-s + (−0.223 − 0.387i)5-s + (0.551 − 0.0625i)6-s + (−0.188 + 0.327i)7-s + 0.939·8-s + (−0.293 + 0.955i)9-s − 0.248·10-s + (0.330 − 0.572i)11-s + (−0.276 + 0.634i)12-s + (0.115 + 0.199i)13-s + (0.104 + 0.181i)14-s + (0.178 − 0.410i)15-s + (−0.0850 + 0.147i)16-s + 0.241·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.839 - 0.543i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(315\)    =    \(3^{2} \cdot 5 \cdot 7\)
Sign: $0.839 - 0.543i$
Analytic conductor: \(2.51528\)
Root analytic conductor: \(1.58596\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{315} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 315,\ (\ :1/2),\ 0.839 - 0.543i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.82960 + 0.540665i\)
\(L(\frac12)\) \(\approx\) \(1.82960 + 0.540665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.02 - 1.39i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.392 + 0.680i)T + (-1 - 1.73i)T^{2} \)
11 \( 1 + (-1.09 + 1.89i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.415 - 0.719i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 0.994T + 17T^{2} \)
19 \( 1 + 2.08T + 19T^{2} \)
23 \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.02 + 5.24i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.857 + 1.48i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 3.89T + 37T^{2} \)
41 \( 1 + (5.67 + 9.83i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-4.85 + 8.41i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.09 + 1.89i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 2.95T + 53T^{2} \)
59 \( 1 + (2.75 + 4.76i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.66 + 6.34i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.60 - 6.24i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 4.51T + 71T^{2} \)
73 \( 1 + 15.1T + 73T^{2} \)
79 \( 1 + (3.91 - 6.77i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.41 - 12.8i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.5T + 89T^{2} \)
97 \( 1 + (5.55 - 9.61i)T + (-48.5 - 84.0i)T^{2} \)
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

−11.72480940507407183760034351391, −10.91103836348147737180229005721, −9.974912184109263569125694846965, −8.810599811031527520211643301151, −8.260394764846553946199262088710, −7.06154947343856845297986050035, −5.52882844206350050434309329092, −4.23025479563775397326988163195, −3.47546807124902971248779218866, −2.23043764810094226181264876317, 1.47684946099260115608005256086, 2.99864511407913339190359926809, 4.49076009788408072394229732319, 5.97795557127560996124585469444, 6.82368129403318477429435230168, 7.42280884666841811775090703747, 8.513277495519568955929706864866, 9.739080969345307897584287912305, 10.62064493186161278094050756733, 11.66130025623705467542815003363

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