↑

Properties

Label 2-3520-3520.109-c0-0-2
Degree $2$
Conductor $3520$
Sign $0.881 - 0.471i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.995 + 0.0980i)2-s + (0.980 + 0.195i)4-s + (−0.195 − 0.980i)5-s + (0.732 + 1.76i)7-s + (0.956 + 0.290i)8-s + (−0.382 + 0.923i)9-s + (−0.0980 − 0.995i)10-s + (0.831 + 0.555i)11-s + (0.301 − 1.51i)13-s + (0.555 + 1.83i)14-s + (0.923 + 0.382i)16-s + (−1.24 − 1.24i)17-s + (−0.471 + 0.881i)18-s − i·20-s + (0.773 + 0.634i)22-s + ⋯
L(s)  = 1  + (0.995 + 0.0980i)2-s + (0.980 + 0.195i)4-s + (−0.195 − 0.980i)5-s + (0.732 + 1.76i)7-s + (0.956 + 0.290i)8-s + (−0.382 + 0.923i)9-s + (−0.0980 − 0.995i)10-s + (0.831 + 0.555i)11-s + (0.301 − 1.51i)13-s + (0.555 + 1.83i)14-s + (0.923 + 0.382i)16-s + (−1.24 − 1.24i)17-s + (−0.471 + 0.881i)18-s − i·20-s + (0.773 + 0.634i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.881 - 0.471i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.881 - 0.471i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.601435498\)
\(L(\frac12)\) \(\approx\) \(2.601435498\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.995 - 0.0980i)T \)
5 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (-0.831 - 0.555i)T \)
good3 \( 1 + (0.382 - 0.923i)T^{2} \)
7 \( 1 + (-0.732 - 1.76i)T + (-0.707 + 0.707i)T^{2} \)
13 \( 1 + (-0.301 + 1.51i)T + (-0.923 - 0.382i)T^{2} \)
17 \( 1 + (1.24 + 1.24i)T + iT^{2} \)
19 \( 1 + (-0.923 - 0.382i)T^{2} \)
23 \( 1 + (-0.707 - 0.707i)T^{2} \)
29 \( 1 + (-0.382 + 0.923i)T^{2} \)
31 \( 1 - 1.66iT - T^{2} \)
37 \( 1 + (-0.923 + 0.382i)T^{2} \)
41 \( 1 + (-0.707 - 0.707i)T^{2} \)
43 \( 1 + (-0.108 + 0.162i)T + (-0.382 - 0.923i)T^{2} \)
47 \( 1 - iT^{2} \)
53 \( 1 + (-0.382 - 0.923i)T^{2} \)
59 \( 1 + (0.0761 + 0.382i)T + (-0.923 + 0.382i)T^{2} \)
61 \( 1 + (0.382 - 0.923i)T^{2} \)
67 \( 1 + (0.382 - 0.923i)T^{2} \)
71 \( 1 + (0.292 + 0.707i)T + (-0.707 + 0.707i)T^{2} \)
73 \( 1 + (-0.761 + 1.83i)T + (-0.707 - 0.707i)T^{2} \)
79 \( 1 + iT^{2} \)
83 \( 1 + (1.51 + 0.301i)T + (0.923 + 0.382i)T^{2} \)
89 \( 1 + (-1.81 + 0.750i)T + (0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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

−8.698456660236016880540285226470, −8.094253842826294662585462706413, −7.37711686330217129128479993195, −6.27700249211773845315516502476, −5.53236870572929339709914824209, −4.94115874820116043056330371458, −4.66789926268490333761915742208, −3.25924735428117736143539929292, −2.41071143050088893879929031807, −1.63153498361365311617763720574, 1.26593223220600358465322014544, 2.28298427539685687831057263973, 3.66512981377428076468809760243, 3.96867465224922216350459503779, 4.41346102630581826064377076664, 5.92278837198886075522493816072, 6.59042947492753187200902422223, 6.82185224057476160063305906341, 7.69076380211568544854096230298, 8.557409270856955577304142449575

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