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Properties

Label 2-312-104.77-c3-0-27
Degree $2$
Conductor $312$
Sign $0.949 - 0.312i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.47 + 1.37i)2-s + 3i·3-s + (4.22 − 6.79i)4-s − 19.0·5-s + (−4.11 − 7.41i)6-s − 2.83i·7-s + (−1.13 + 22.5i)8-s − 9·9-s + (47.0 − 26.1i)10-s − 28.0·11-s + (20.3 + 12.6i)12-s + (−43.7 + 16.8i)13-s + (3.88 + 7.00i)14-s − 57.1i·15-s + (−28.2 − 57.4i)16-s − 86.5·17-s + ⋯
L(s)  = 1  + (−0.874 + 0.485i)2-s + 0.577i·3-s + (0.528 − 0.848i)4-s − 1.70·5-s + (−0.280 − 0.504i)6-s − 0.152i·7-s + (−0.0500 + 0.998i)8-s − 0.333·9-s + (1.48 − 0.826i)10-s − 0.769·11-s + (0.490 + 0.305i)12-s + (−0.932 + 0.359i)13-s + (0.0742 + 0.133i)14-s − 0.983i·15-s + (−0.441 − 0.897i)16-s − 1.23·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.949 - 0.312i$
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: \((2,\ 312,\ (\ :3/2),\ 0.949 - 0.312i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.4543935193\)
\(L(\frac12)\) \(\approx\) \(0.4543935193\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.47 - 1.37i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (43.7 - 16.8i)T \)
good5 \( 1 + 19.0T + 125T^{2} \)
7 \( 1 + 2.83iT - 343T^{2} \)
11 \( 1 + 28.0T + 1.33e3T^{2} \)
17 \( 1 + 86.5T + 4.91e3T^{2} \)
19 \( 1 - 82.3T + 6.85e3T^{2} \)
23 \( 1 - 81.3T + 1.21e4T^{2} \)
29 \( 1 + 70.2iT - 2.43e4T^{2} \)
31 \( 1 + 184. iT - 2.97e4T^{2} \)
37 \( 1 + 62.8T + 5.06e4T^{2} \)
41 \( 1 - 405. iT - 6.89e4T^{2} \)
43 \( 1 - 257. iT - 7.95e4T^{2} \)
47 \( 1 + 200. iT - 1.03e5T^{2} \)
53 \( 1 + 121. iT - 1.48e5T^{2} \)
59 \( 1 - 513.T + 2.05e5T^{2} \)
61 \( 1 + 54.0iT - 2.26e5T^{2} \)
67 \( 1 - 438.T + 3.00e5T^{2} \)
71 \( 1 - 181. iT - 3.57e5T^{2} \)
73 \( 1 - 922. iT - 3.89e5T^{2} \)
79 \( 1 - 158.T + 4.93e5T^{2} \)
83 \( 1 - 324.T + 5.71e5T^{2} \)
89 \( 1 + 990. iT - 7.04e5T^{2} \)
97 \( 1 + 1.57e3iT - 9.12e5T^{2} \)
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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

−11.39951980176477108833604768396, −10.26294596001922415300393272690, −9.339849943199741063970762510381, −8.326093203131992460041724513451, −7.60938292071686077230033248190, −6.81411218702690322558261216671, −5.18942918397386235296000839859, −4.27137747027636784071014119426, −2.72586054016691957485573958994, −0.41139479749537998394453127304, 0.62025494560991079958149125553, 2.49920983894444725647533911904, 3.57628785435341489485905052726, 5.01502258383827764468976900378, 7.01603065182728898890187761166, 7.40531482702291046983105524131, 8.328150178324247334644113117409, 9.081477955479122869362563738324, 10.53315690557968710730775270464, 11.15644847177377622120344243000

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