Properties

Label 2-912-57.41-c1-0-31
Degree $2$
Conductor $912$
Sign $0.286 + 0.958i$
Analytic cond. $7.28235$
Root an. cond. $2.69858$
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  + (0.592 + 1.62i)3-s + (−2.47 − 4.28i)7-s + (−2.29 + 1.92i)9-s + (1.08 − 2.98i)13-s + (−0.5 − 4.33i)19-s + (5.50 − 6.55i)21-s + (−4.69 − 1.71i)25-s + (−4.5 − 2.59i)27-s + (4.66 − 2.69i)31-s − 11.7i·37-s + 5.49·39-s + (0.0957 − 0.543i)43-s + (−8.71 + 15.1i)49-s + (6.75 − 3.37i)57-s + (0.762 + 4.32i)61-s + ⋯
L(s)  = 1  + (0.342 + 0.939i)3-s + (−0.934 − 1.61i)7-s + (−0.766 + 0.642i)9-s + (0.300 − 0.826i)13-s + (−0.114 − 0.993i)19-s + (1.20 − 1.43i)21-s + (−0.939 − 0.342i)25-s + (−0.866 − 0.499i)27-s + (0.838 − 0.484i)31-s − 1.92i·37-s + 0.879·39-s + (0.0146 − 0.0828i)43-s + (−1.24 + 2.15i)49-s + (0.894 − 0.447i)57-s + (0.0976 + 0.553i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.286 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(912\)    =    \(2^{4} \cdot 3 \cdot 19\)
Sign: $0.286 + 0.958i$
Analytic conductor: \(7.28235\)
Root analytic conductor: \(2.69858\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{912} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 912,\ (\ :1/2),\ 0.286 + 0.958i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.883444 - 0.658209i\)
\(L(\frac12)\) \(\approx\) \(0.883444 - 0.658209i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.592 - 1.62i)T \)
19 \( 1 + (0.5 + 4.33i)T \)
good5 \( 1 + (4.69 + 1.71i)T^{2} \)
7 \( 1 + (2.47 + 4.28i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.08 + 2.98i)T + (-9.95 - 8.35i)T^{2} \)
17 \( 1 + (-2.95 - 16.7i)T^{2} \)
23 \( 1 + (21.6 - 7.86i)T^{2} \)
29 \( 1 + (5.03 - 28.5i)T^{2} \)
31 \( 1 + (-4.66 + 2.69i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 11.7iT - 37T^{2} \)
41 \( 1 + (31.4 - 26.3i)T^{2} \)
43 \( 1 + (-0.0957 + 0.543i)T + (-40.4 - 14.7i)T^{2} \)
47 \( 1 + (-8.16 + 46.2i)T^{2} \)
53 \( 1 + (-49.8 + 18.1i)T^{2} \)
59 \( 1 + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-0.762 - 4.32i)T + (-57.3 + 20.8i)T^{2} \)
67 \( 1 + (-1.42 - 1.69i)T + (-11.6 + 65.9i)T^{2} \)
71 \( 1 + (-66.7 - 24.2i)T^{2} \)
73 \( 1 + (-14.4 + 5.26i)T + (55.9 - 46.9i)T^{2} \)
79 \( 1 + (-5.09 - 14.0i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (68.1 + 57.2i)T^{2} \)
97 \( 1 + (3.34 - 3.98i)T + (-16.8 - 95.5i)T^{2} \)
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

−9.881512659784445312204333808307, −9.379735783572519491400951492973, −8.243072511662298424610726649249, −7.47420757709916163332268993789, −6.51049268056874839778112033554, −5.45901460375325237500302622062, −4.26070063880055380678030553923, −3.71301670186060922436333326105, −2.66223855023195415361548740522, −0.48694326412182266934276609103, 1.69017078653650306404017444731, 2.69233575885012420473105664725, 3.63760929399028963353355971717, 5.26064188261127967646138343871, 6.26310385656869515477935385730, 6.55634735269529140392997718882, 7.889406976582373192816151685303, 8.564949041518089805754635759787, 9.296321470081883441528010339347, 9.984664211648505456465607443119

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