↑

Properties

Label 2-93-31.16-c1-0-0
Degree $2$
Conductor $93$
Sign $-0.919 - 0.392i$
Analytic cond. $0.742608$
Root an. cond. $0.861747$
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  + (−0.968 + 0.703i)2-s + (−0.809 − 0.587i)3-s + (−0.175 + 0.538i)4-s − 2.70·5-s + 1.19·6-s + (−1.33 + 4.11i)7-s + (−0.949 − 2.92i)8-s + (0.309 + 0.951i)9-s + (2.61 − 1.90i)10-s + (−0.299 + 0.920i)11-s + (0.458 − 0.332i)12-s + (−2.55 − 1.85i)13-s + (−1.59 − 4.92i)14-s + (2.18 + 1.58i)15-s + (2.06 + 1.49i)16-s + (1.52 + 4.68i)17-s + ⋯
L(s)  = 1  + (−0.684 + 0.497i)2-s + (−0.467 − 0.339i)3-s + (−0.0875 + 0.269i)4-s − 1.20·5-s + 0.488·6-s + (−0.504 + 1.55i)7-s + (−0.335 − 1.03i)8-s + (0.103 + 0.317i)9-s + (0.828 − 0.601i)10-s + (−0.0901 + 0.277i)11-s + (0.132 − 0.0960i)12-s + (−0.707 − 0.513i)13-s + (−0.427 − 1.31i)14-s + (0.564 + 0.410i)15-s + (0.515 + 0.374i)16-s + (0.369 + 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(93\)    =    \(3 \cdot 31\)
Sign: $-0.919 - 0.392i$
Analytic conductor: \(0.742608\)
Root analytic conductor: \(0.861747\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{93} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 93,\ (\ :1/2),\ -0.919 - 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0584991 + 0.286450i\)
\(L(\frac12)\) \(\approx\) \(0.0584991 + 0.286450i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (-5.08 - 2.25i)T \)
good2 \( 1 + (0.968 - 0.703i)T + (0.618 - 1.90i)T^{2} \)
5 \( 1 + 2.70T + 5T^{2} \)
7 \( 1 + (1.33 - 4.11i)T + (-5.66 - 4.11i)T^{2} \)
11 \( 1 + (0.299 - 0.920i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (2.55 + 1.85i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-1.52 - 4.68i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-4.82 + 3.50i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (-1.05 - 3.24i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (4.12 - 2.99i)T + (8.96 - 27.5i)T^{2} \)
37 \( 1 + 8.13T + 37T^{2} \)
41 \( 1 + (2.61 - 1.90i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + (4.94 - 3.59i)T + (13.2 - 40.8i)T^{2} \)
47 \( 1 + (6.41 + 4.65i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.66 - 5.11i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-11.0 - 8.00i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 - 1.99T + 61T^{2} \)
67 \( 1 - 7.01T + 67T^{2} \)
71 \( 1 + (1.95 + 6.00i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-0.0487 + 0.149i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-5.17 - 15.9i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-5.63 + 4.09i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + (-2.78 + 8.56i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-0.516 + 1.58i)T + (-78.4 - 57.0i)T^{2} \)
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

−15.14406161569452408065742849159, −13.04630140538560847876612290830, −12.24067672963173359007862515950, −11.66876234241400982492707151255, −9.928056854509285513310169660830, −8.741036241464453885062158369144, −7.83592864632838951925469316395, −6.83137157374510955113694369409, −5.31368509094287178317764582355, −3.29989935687544110627254859501, 0.46176326705841467664521555369, 3.62180501773907674527975367246, 5.00619016605532570452336588010, 6.93686943413545166917723169266, 7.999911942652832037201383504364, 9.614508155412973703118875399376, 10.25611027484439163539457115793, 11.36177103390638363114331121923, 11.98344104311595029055408223127, 13.67445829467366017083338985979

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