↑

Properties

Label 2-312-104.77-c3-0-32
Degree $2$
Conductor $312$
Sign $-0.0966 - 0.995i$
Analytic cond. $18.4085$
Root an. cond. $4.29052$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.98 + 2.01i)2-s + 3i·3-s + (−0.145 − 7.99i)4-s + 19.4·5-s + (−6.05 − 5.94i)6-s − 8.53i·7-s + (16.4 + 15.5i)8-s − 9·9-s + (−38.5 + 39.2i)10-s − 49.3·11-s + (23.9 − 0.436i)12-s + (−30.7 + 35.3i)13-s + (17.2 + 16.9i)14-s + 58.2i·15-s + (−63.9 + 2.32i)16-s + 117.·17-s + ⋯
L(s)  = 1  + (−0.700 + 0.713i)2-s + 0.577i·3-s + (−0.0181 − 0.999i)4-s + 1.73·5-s + (−0.411 − 0.404i)6-s − 0.460i·7-s + (0.726 + 0.687i)8-s − 0.333·9-s + (−1.21 + 1.24i)10-s − 1.35·11-s + (0.577 − 0.0104i)12-s + (−0.656 + 0.754i)13-s + (0.328 + 0.322i)14-s + 1.00i·15-s + (−0.999 + 0.0363i)16-s + 1.67·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0966 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.0966 - 0.995i$
Analytic conductor: \(18.4085\)
Root analytic conductor: \(4.29052\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (181, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :3/2),\ -0.0966 - 0.995i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.625998429\)
\(L(\frac12)\) \(\approx\) \(1.625998429\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.98 - 2.01i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (30.7 - 35.3i)T \)
good5 \( 1 - 19.4T + 125T^{2} \)
7 \( 1 + 8.53iT - 343T^{2} \)
11 \( 1 + 49.3T + 1.33e3T^{2} \)
17 \( 1 - 117.T + 4.91e3T^{2} \)
19 \( 1 - 26.5T + 6.85e3T^{2} \)
23 \( 1 - 117.T + 1.21e4T^{2} \)
29 \( 1 - 222. iT - 2.43e4T^{2} \)
31 \( 1 - 240. iT - 2.97e4T^{2} \)
37 \( 1 - 192.T + 5.06e4T^{2} \)
41 \( 1 + 4.97iT - 6.89e4T^{2} \)
43 \( 1 - 184. iT - 7.95e4T^{2} \)
47 \( 1 + 410. iT - 1.03e5T^{2} \)
53 \( 1 - 501. iT - 1.48e5T^{2} \)
59 \( 1 + 353.T + 2.05e5T^{2} \)
61 \( 1 + 268. iT - 2.26e5T^{2} \)
67 \( 1 - 623.T + 3.00e5T^{2} \)
71 \( 1 + 313. iT - 3.57e5T^{2} \)
73 \( 1 - 254. iT - 3.89e5T^{2} \)
79 \( 1 - 866.T + 4.93e5T^{2} \)
83 \( 1 + 901.T + 5.71e5T^{2} \)
89 \( 1 - 701. iT - 7.04e5T^{2} \)
97 \( 1 - 531. iT - 9.12e5T^{2} \)
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

−10.84613179085647323206478518957, −10.28220415423339957547651855093, −9.658121795261625306598626426385, −8.905502088659096942446191582316, −7.62534829571809221414858940425, −6.65970680584818620298851897616, −5.38965448427754564485884585625, −5.07968049000033226434953812066, −2.77623915075750906937957911914, −1.29511750649894900055775075103, 0.838785801215577330135936578624, 2.27440959732291163605150805336, 2.87076579484850844144756773160, 5.20697236100635569810167651402, 5.95383435502165232597141092799, 7.44509848009414898925810199506, 8.154294314223317523771442805050, 9.486701936480599324580159769734, 9.913539995138962051326913916535, 10.77039814440059133096486136599

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