Properties

Label 2-546-39.32-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.926 - 0.376i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (−1.67 + 0.454i)3-s + (0.866 − 0.499i)4-s + (−0.359 − 0.359i)5-s + (1.49 − 0.871i)6-s + (−0.258 + 0.965i)7-s + (−0.707 + 0.707i)8-s + (2.58 − 1.51i)9-s + (0.440 + 0.254i)10-s + (−0.529 − 1.97i)11-s + (−1.22 + 1.22i)12-s + (−0.322 + 3.59i)13-s i·14-s + (0.763 + 0.437i)15-s + (0.500 − 0.866i)16-s + (−0.334 − 0.579i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (−0.964 + 0.262i)3-s + (0.433 − 0.249i)4-s + (−0.160 − 0.160i)5-s + (0.611 − 0.355i)6-s + (−0.0978 + 0.365i)7-s + (−0.249 + 0.249i)8-s + (0.862 − 0.506i)9-s + (0.139 + 0.0803i)10-s + (−0.159 − 0.595i)11-s + (−0.352 + 0.354i)12-s + (−0.0895 + 0.995i)13-s − 0.267i·14-s + (0.197 + 0.112i)15-s + (0.125 − 0.216i)16-s + (−0.0810 − 0.140i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.926 - 0.376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.926 - 0.376i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.926 - 0.376i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0529431 + 0.270640i\)
\(L(\frac12)\) \(\approx\) \(0.0529431 + 0.270640i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.965 - 0.258i)T \)
3 \( 1 + (1.67 - 0.454i)T \)
7 \( 1 + (0.258 - 0.965i)T \)
13 \( 1 + (0.322 - 3.59i)T \)
good5 \( 1 + (0.359 + 0.359i)T + 5iT^{2} \)
11 \( 1 + (0.529 + 1.97i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.334 + 0.579i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-3.11 - 0.834i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (3.54 - 6.14i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.59 + 1.49i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.15 - 4.15i)T - 31iT^{2} \)
37 \( 1 + (4.74 - 1.27i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (5.56 - 1.49i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (4.09 - 2.36i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (9.15 - 9.15i)T - 47iT^{2} \)
53 \( 1 + 2.70iT - 53T^{2} \)
59 \( 1 + (-1.53 - 0.411i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-1.05 - 1.83i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.55 + 13.2i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (3.72 - 13.9i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.50 - 2.50i)T + 73iT^{2} \)
79 \( 1 + 14.0T + 79T^{2} \)
83 \( 1 + (2.91 + 2.91i)T + 83iT^{2} \)
89 \( 1 + (-3.28 - 12.2i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-16.9 - 4.54i)T + (84.0 + 48.5i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.35357388913386280014286181331, −10.16614017818374291167241631765, −9.559251145648623427236912211416, −8.642823239909781225771792766404, −7.57445437451289613472539626037, −6.61560750039990711884049337157, −5.78509532231813344027027088851, −4.85147022244153119670451938319, −3.49300821605133391841492798345, −1.60184242821942498511918798959, 0.22498388075535700587607705678, 1.85919938768563560488414256794, 3.49404007565140960151558259863, 4.88693134698911329472745613210, 5.88899315603065500970934334731, 7.03846365356044160664160204964, 7.51979016879126139196372116833, 8.601758893060546411199490564404, 9.929140020274493773289603839899, 10.33038367691229981662044114507

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