↑

Properties

Label 2-2352-7.2-c1-0-3
Degree $2$
Conductor $2352$
Sign $-0.386 - 0.922i$
Analytic cond. $18.7808$
Root an. cond. $4.33368$
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  + (−0.5 − 0.866i)3-s + (−0.5 + 0.866i)5-s + (−0.499 + 0.866i)9-s + (1.5 + 2.59i)11-s − 4·13-s + 0.999·15-s + (2 − 3.46i)19-s + (4 − 6.92i)23-s + (2 + 3.46i)25-s + 0.999·27-s − 3·29-s + (2.5 + 4.33i)31-s + (1.5 − 2.59i)33-s + (−4 + 6.92i)37-s + (2 + 3.46i)39-s + ⋯
L(s)  = 1  + (−0.288 − 0.499i)3-s + (−0.223 + 0.387i)5-s + (−0.166 + 0.288i)9-s + (0.452 + 0.783i)11-s − 1.10·13-s + 0.258·15-s + (0.458 − 0.794i)19-s + (0.834 − 1.44i)23-s + (0.400 + 0.692i)25-s + 0.192·27-s − 0.557·29-s + (0.449 + 0.777i)31-s + (0.261 − 0.452i)33-s + (−0.657 + 1.13i)37-s + (0.320 + 0.554i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2352 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2352\)    =    \(2^{4} \cdot 3 \cdot 7^{2}\)
Sign: $-0.386 - 0.922i$
Analytic conductor: \(18.7808\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2352} (961, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2352,\ (\ :1/2),\ -0.386 - 0.922i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7092918257\)
\(L(\frac12)\) \(\approx\) \(0.7092918257\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 \)
good5 \( 1 + (0.5 - 0.866i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-4 + 6.92i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 3T + 29T^{2} \)
31 \( 1 + (-2.5 - 4.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (4 - 6.92i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + 8T + 41T^{2} \)
43 \( 1 + 6T + 43T^{2} \)
47 \( 1 + (5 - 8.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.5 + 7.79i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.5 - 4.33i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (5 - 8.66i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3 - 5.19i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 10T + 71T^{2} \)
73 \( 1 + (-1 - 1.73i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-5.5 + 9.52i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7T + 83T^{2} \)
89 \( 1 + (9 - 15.5i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 17T + 97T^{2} \)
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

−9.194876064339842003568239904173, −8.425413564905511154987214016351, −7.43988780686737350427156232004, −6.92287130462600519117958026148, −6.43811849957182432265936726686, −4.99713947125885260120795000339, −4.78726648717633100988353356955, −3.33031850749706265711882217311, −2.50654297864517688273665628416, −1.31647440626312427738070601903, 0.25885663413290007122068123422, 1.69105443259702635409753867140, 3.14689938647144889625323465785, 3.81961643616638987090751789741, 4.89347479247609446234642688416, 5.41459837996917741216351696117, 6.33173236436501707085106142303, 7.24610891999449285023194784912, 8.019908812639863456147524437632, 8.835846931275869296727525337777

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