↑

Properties

Label 2-3520-3520.3189-c0-0-2
Degree $2$
Conductor $3520$
Sign $-0.471 + 0.881i$
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  + (0.290 − 0.956i)2-s + (−0.831 − 0.555i)4-s + (−0.555 − 0.831i)5-s + (1.42 + 0.591i)7-s + (−0.773 + 0.634i)8-s + (0.923 − 0.382i)9-s + (−0.956 + 0.290i)10-s + (0.195 − 0.980i)11-s + (−0.979 + 1.46i)13-s + (0.980 − 1.19i)14-s + (0.382 + 0.923i)16-s + (1.40 − 1.40i)17-s + (−0.0980 − 0.995i)18-s + i·20-s + (−0.881 − 0.471i)22-s + ⋯
L(s)  = 1  + (0.290 − 0.956i)2-s + (−0.831 − 0.555i)4-s + (−0.555 − 0.831i)5-s + (1.42 + 0.591i)7-s + (−0.773 + 0.634i)8-s + (0.923 − 0.382i)9-s + (−0.956 + 0.290i)10-s + (0.195 − 0.980i)11-s + (−0.979 + 1.46i)13-s + (0.980 − 1.19i)14-s + (0.382 + 0.923i)16-s + (1.40 − 1.40i)17-s + (−0.0980 − 0.995i)18-s + i·20-s + (−0.881 − 0.471i)22-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $-0.471 + 0.881i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (3189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ -0.471 + 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.518351506\)
\(L(\frac12)\) \(\approx\) \(1.518351506\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.290 + 0.956i)T \)
5 \( 1 + (0.555 + 0.831i)T \)
11 \( 1 + (-0.195 + 0.980i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
7 \( 1 + (-1.42 - 0.591i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.979 - 1.46i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (-1.40 + 1.40i)T - iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 + 0.390iT - T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (-1.87 - 0.373i)T + (0.923 + 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (0.536 - 0.222i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (1.46 + 0.979i)T + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (0.636 - 1.53i)T + (-0.707 - 0.707i)T^{2} \)
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

−8.747987267017494333828667699553, −7.86610553211875280083820563500, −7.27543645558302629704253034399, −5.92191785023745217996165946382, −5.14256277553974143546112838611, −4.58872251187742404535650213894, −4.01559417404522890617995181115, −2.88208671679790865824818871617, −1.76744179470362924865968200979, −0.985741601710553201000349815830, 1.38677335803672208760052905815, 2.79069900357607298701905792999, 3.94157465942124110351776816815, 4.39941601958925199725452594704, 5.20897575145180905674399514276, 5.96549441047255444004925163908, 7.19761386964145686573787537961, 7.48493478253225515115841713669, 7.82465799321115839222999072801, 8.566853972544857881102918778582

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