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Properties

Label 2-312-104.77-c3-0-18
Degree $2$
Conductor $312$
Sign $-0.997 - 0.0679i$
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  + (1.61 + 2.32i)2-s − 3i·3-s + (−2.78 + 7.50i)4-s − 1.96·5-s + (6.96 − 4.84i)6-s + 7.12i·7-s + (−21.9 + 5.66i)8-s − 9·9-s + (−3.16 − 4.55i)10-s + 39.6·11-s + (22.5 + 8.34i)12-s + (−14.7 + 44.4i)13-s + (−16.5 + 11.5i)14-s + 5.88i·15-s + (−48.5 − 41.7i)16-s − 20.3·17-s + ⋯
L(s)  = 1  + (0.571 + 0.820i)2-s − 0.577i·3-s + (−0.347 + 0.937i)4-s − 0.175·5-s + (0.473 − 0.329i)6-s + 0.384i·7-s + (−0.968 + 0.250i)8-s − 0.333·9-s + (−0.100 − 0.144i)10-s + 1.08·11-s + (0.541 + 0.200i)12-s + (−0.315 + 0.948i)13-s + (−0.315 + 0.219i)14-s + 0.101i·15-s + (−0.758 − 0.651i)16-s − 0.291·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0679i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0679i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.997 - 0.0679i$
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: \((2,\ 312,\ (\ :3/2),\ -0.997 - 0.0679i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.175106443\)
\(L(\frac12)\) \(\approx\) \(1.175106443\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.61 - 2.32i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (14.7 - 44.4i)T \)
good5 \( 1 + 1.96T + 125T^{2} \)
7 \( 1 - 7.12iT - 343T^{2} \)
11 \( 1 - 39.6T + 1.33e3T^{2} \)
17 \( 1 + 20.3T + 4.91e3T^{2} \)
19 \( 1 + 78.7T + 6.85e3T^{2} \)
23 \( 1 + 108.T + 1.21e4T^{2} \)
29 \( 1 - 306. iT - 2.43e4T^{2} \)
31 \( 1 + 122. iT - 2.97e4T^{2} \)
37 \( 1 + 238.T + 5.06e4T^{2} \)
41 \( 1 + 113. iT - 6.89e4T^{2} \)
43 \( 1 - 443. iT - 7.95e4T^{2} \)
47 \( 1 - 435. iT - 1.03e5T^{2} \)
53 \( 1 + 496. iT - 1.48e5T^{2} \)
59 \( 1 + 868.T + 2.05e5T^{2} \)
61 \( 1 - 355. iT - 2.26e5T^{2} \)
67 \( 1 - 792.T + 3.00e5T^{2} \)
71 \( 1 - 208. iT - 3.57e5T^{2} \)
73 \( 1 + 1.07e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 - 701.T + 5.71e5T^{2} \)
89 \( 1 + 14.2iT - 7.04e5T^{2} \)
97 \( 1 + 1.16e3iT - 9.12e5T^{2} \)
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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

−12.05802638365221263600142893425, −11.13384368619659554796744603825, −9.413195573944169466435412668532, −8.708617659263362759983871252433, −7.68590293090423684488669695757, −6.68623001322530630681395252530, −6.07383643613112503615604336790, −4.69426634241820662588612337163, −3.64471917014976640182340188897, −2.00296231675693894002412812008, 0.33717267142077076445820872170, 2.12682803628114191530352427063, 3.65618098516810368778752284135, 4.28443628418535902414698655966, 5.55347705743131396151837565521, 6.56552204624418173304374340444, 8.121812880230532038973822059267, 9.220448911541678364082195969229, 10.11716548853408313609085389302, 10.73847266888425848734763714460

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