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Properties

Label 2-312-104.77-c3-0-29
Degree $2$
Conductor $312$
Sign $-0.213 - 0.976i$
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  + (−1.33 + 2.49i)2-s − 3i·3-s + (−4.41 − 6.67i)4-s + 12.3·5-s + (7.47 + 4.01i)6-s + 26.6i·7-s + (22.5 − 2.06i)8-s − 9·9-s + (−16.5 + 30.8i)10-s − 33.2·11-s + (−20.0 + 13.2i)12-s + (46.5 − 5.78i)13-s + (−66.3 − 35.6i)14-s − 37.1i·15-s + (−25.0 + 58.8i)16-s + 16.3·17-s + ⋯
L(s)  = 1  + (−0.473 + 0.880i)2-s − 0.577i·3-s + (−0.551 − 0.834i)4-s + 1.10·5-s + (0.508 + 0.273i)6-s + 1.43i·7-s + (0.995 − 0.0911i)8-s − 0.333·9-s + (−0.523 + 0.974i)10-s − 0.911·11-s + (−0.481 + 0.318i)12-s + (0.992 − 0.123i)13-s + (−1.26 − 0.680i)14-s − 0.638i·15-s + (−0.391 + 0.920i)16-s + 0.232·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.213 - 0.976i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.213 - 0.976i$
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: \((2,\ 312,\ (\ :3/2),\ -0.213 - 0.976i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.444865151\)
\(L(\frac12)\) \(\approx\) \(1.444865151\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.33 - 2.49i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (-46.5 + 5.78i)T \)
good5 \( 1 - 12.3T + 125T^{2} \)
7 \( 1 - 26.6iT - 343T^{2} \)
11 \( 1 + 33.2T + 1.33e3T^{2} \)
17 \( 1 - 16.3T + 4.91e3T^{2} \)
19 \( 1 + 43.9T + 6.85e3T^{2} \)
23 \( 1 - 60.7T + 1.21e4T^{2} \)
29 \( 1 - 218. iT - 2.43e4T^{2} \)
31 \( 1 - 12.9iT - 2.97e4T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 - 241. iT - 6.89e4T^{2} \)
43 \( 1 - 290. iT - 7.95e4T^{2} \)
47 \( 1 - 385. iT - 1.03e5T^{2} \)
53 \( 1 + 220. iT - 1.48e5T^{2} \)
59 \( 1 - 310.T + 2.05e5T^{2} \)
61 \( 1 - 156. iT - 2.26e5T^{2} \)
67 \( 1 + 586.T + 3.00e5T^{2} \)
71 \( 1 - 1.14e3iT - 3.57e5T^{2} \)
73 \( 1 + 792. iT - 3.89e5T^{2} \)
79 \( 1 + 1.06e3T + 4.93e5T^{2} \)
83 \( 1 - 1.48e3T + 5.71e5T^{2} \)
89 \( 1 - 216. iT - 7.04e5T^{2} \)
97 \( 1 - 789. iT - 9.12e5T^{2} \)
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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

−11.36429788779593588271429192996, −10.36491626838017630915639789110, −9.318160004855737600442749965093, −8.661975184091095532975861290924, −7.78219507055084595203912404113, −6.39697787492149122262957754639, −5.86940775746214712961721036347, −5.04556883808092908467625052267, −2.66379590297954384133863008641, −1.40580150503090330471506532418, 0.66204714787393980638487822392, 2.16319334727392126300187008625, 3.57589525370663841855502259461, 4.56100936331362240759740445698, 5.90037802704395272039271894943, 7.33276290001017693370602897220, 8.388493182348829083579717730408, 9.421922665909396306444933004387, 10.29682239921751268249073651615, 10.58549328893972429471539680105

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