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Properties

Label 2-312-104.77-c3-0-53
Degree $2$
Conductor $312$
Sign $0.233 + 0.972i$
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  + (−2.73 + 0.722i)2-s + 3i·3-s + (6.95 − 3.95i)4-s − 2.18·5-s + (−2.16 − 8.20i)6-s + 4.02i·7-s + (−16.1 + 15.8i)8-s − 9·9-s + (5.98 − 1.58i)10-s + 8.07·11-s + (11.8 + 20.8i)12-s + (−40.2 − 24.0i)13-s + (−2.91 − 11.0i)14-s − 6.56i·15-s + (32.7 − 54.9i)16-s + 130.·17-s + ⋯
L(s)  = 1  + (−0.966 + 0.255i)2-s + 0.577i·3-s + (0.869 − 0.493i)4-s − 0.195·5-s + (−0.147 − 0.558i)6-s + 0.217i·7-s + (−0.714 + 0.699i)8-s − 0.333·9-s + (0.189 − 0.0500i)10-s + 0.221·11-s + (0.285 + 0.501i)12-s + (−0.858 − 0.513i)13-s + (−0.0555 − 0.210i)14-s − 0.113i·15-s + (0.511 − 0.858i)16-s + 1.85·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.233 + 0.972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.233 + 0.972i$
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: \((2,\ 312,\ (\ :3/2),\ 0.233 + 0.972i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.5057728755\)
\(L(\frac12)\) \(\approx\) \(0.5057728755\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.73 - 0.722i)T \)
3 \( 1 - 3iT \)
13 \( 1 + (40.2 + 24.0i)T \)
good5 \( 1 + 2.18T + 125T^{2} \)
7 \( 1 - 4.02iT - 343T^{2} \)
11 \( 1 - 8.07T + 1.33e3T^{2} \)
17 \( 1 - 130.T + 4.91e3T^{2} \)
19 \( 1 + 109.T + 6.85e3T^{2} \)
23 \( 1 + 77.1T + 1.21e4T^{2} \)
29 \( 1 - 94.2iT - 2.43e4T^{2} \)
31 \( 1 + 171. iT - 2.97e4T^{2} \)
37 \( 1 + 18.4T + 5.06e4T^{2} \)
41 \( 1 + 345. iT - 6.89e4T^{2} \)
43 \( 1 + 272. iT - 7.95e4T^{2} \)
47 \( 1 - 278. iT - 1.03e5T^{2} \)
53 \( 1 + 443. iT - 1.48e5T^{2} \)
59 \( 1 - 25.5T + 2.05e5T^{2} \)
61 \( 1 + 606. iT - 2.26e5T^{2} \)
67 \( 1 - 583.T + 3.00e5T^{2} \)
71 \( 1 + 847. iT - 3.57e5T^{2} \)
73 \( 1 + 1.20e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.18e3T + 4.93e5T^{2} \)
83 \( 1 - 932.T + 5.71e5T^{2} \)
89 \( 1 - 227. iT - 7.04e5T^{2} \)
97 \( 1 - 710. iT - 9.12e5T^{2} \)
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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

−10.69383755996300042924077967097, −10.04602160904693767532863053039, −9.265295010923301934939047069178, −8.183992930571683403368302867536, −7.49593068611087545293317358230, −6.13683384574623218531761082819, −5.24577340348514537064626929094, −3.63109986874521961763994052248, −2.15605884907031145629394991764, −0.26507450132968055833835070411, 1.27753929219485460173474617584, 2.57720979129849741517076338613, 4.00805413017369439723361059240, 5.82793920420268244745871117088, 6.89716958262368666752914537289, 7.73999328625144500869022340411, 8.471667313436460523285118533573, 9.690304628267356136253143475353, 10.31310015996130299565043076899, 11.55709450742505774821753384316

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