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Properties

Label 2-312-104.77-c3-0-15
Degree $2$
Conductor $312$
Sign $0.535 - 0.844i$
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  + (0.457 − 2.79i)2-s + 3i·3-s + (−7.58 − 2.55i)4-s − 4.56·5-s + (8.37 + 1.37i)6-s − 23.0i·7-s + (−10.5 + 19.9i)8-s − 9·9-s + (−2.08 + 12.7i)10-s + 9.02·11-s + (7.65 − 22.7i)12-s + (−3.63 + 46.7i)13-s + (−64.1 − 10.5i)14-s − 13.7i·15-s + (50.9 + 38.7i)16-s − 59.4·17-s + ⋯
L(s)  = 1  + (0.161 − 0.986i)2-s + 0.577i·3-s + (−0.947 − 0.319i)4-s − 0.408·5-s + (0.569 + 0.0933i)6-s − 1.24i·7-s + (−0.468 + 0.883i)8-s − 0.333·9-s + (−0.0660 + 0.403i)10-s + 0.247·11-s + (0.184 − 0.547i)12-s + (−0.0776 + 0.996i)13-s + (−1.22 − 0.200i)14-s − 0.235i·15-s + (0.796 + 0.604i)16-s − 0.847·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.535 - 0.844i$
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: \((2,\ 312,\ (\ :3/2),\ 0.535 - 0.844i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.7884617353\)
\(L(\frac12)\) \(\approx\) \(0.7884617353\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.457 + 2.79i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (3.63 - 46.7i)T \)
good5 \( 1 + 4.56T + 125T^{2} \)
7 \( 1 + 23.0iT - 343T^{2} \)
11 \( 1 - 9.02T + 1.33e3T^{2} \)
17 \( 1 + 59.4T + 4.91e3T^{2} \)
19 \( 1 - 27.0T + 6.85e3T^{2} \)
23 \( 1 - 35.7T + 1.21e4T^{2} \)
29 \( 1 - 142. iT - 2.43e4T^{2} \)
31 \( 1 - 303. iT - 2.97e4T^{2} \)
37 \( 1 - 241.T + 5.06e4T^{2} \)
41 \( 1 - 280. iT - 6.89e4T^{2} \)
43 \( 1 - 198. iT - 7.95e4T^{2} \)
47 \( 1 - 319. iT - 1.03e5T^{2} \)
53 \( 1 - 531. iT - 1.48e5T^{2} \)
59 \( 1 + 378.T + 2.05e5T^{2} \)
61 \( 1 + 878. iT - 2.26e5T^{2} \)
67 \( 1 + 247.T + 3.00e5T^{2} \)
71 \( 1 - 182. iT - 3.57e5T^{2} \)
73 \( 1 + 815. iT - 3.89e5T^{2} \)
79 \( 1 + 817.T + 4.93e5T^{2} \)
83 \( 1 - 246.T + 5.71e5T^{2} \)
89 \( 1 + 355. iT - 7.04e5T^{2} \)
97 \( 1 + 506. iT - 9.12e5T^{2} \)
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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

−11.18678390501496189503485160599, −10.70378961851344839449185072317, −9.654055871443995520430688663705, −8.945505394839683755568505899850, −7.70344500720104426749594598967, −6.44282991343427040395004563343, −4.77473899967453452036849147174, −4.18733610341377570301551111224, −3.13143757850359459782754126303, −1.35130159795524443482961667975, 0.29403411211544446040597298569, 2.52109064090467456732105108831, 4.02826198240198967874517905953, 5.45332511841025867941507452630, 6.07981520439377211280172808282, 7.27345899462893927856996352862, 8.108453037323371702892475800057, 8.865296127476234615376738363802, 9.837425290656354022334400689430, 11.44178039231807409559943673376

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