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Properties

Label 2-312-104.77-c3-0-78
Degree $2$
Conductor $312$
Sign $-0.555 - 0.831i$
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  + (−2.23 + 1.73i)2-s − 3i·3-s + (1.96 − 7.75i)4-s − 12.2·5-s + (5.21 + 6.69i)6-s − 32.4i·7-s + (9.07 + 20.7i)8-s − 9·9-s + (27.4 − 21.3i)10-s − 68.1·11-s + (−23.2 − 5.89i)12-s + (−8.22 − 46.1i)13-s + (56.2 + 72.3i)14-s + 36.8i·15-s + (−56.2 − 30.4i)16-s + 86.0·17-s + ⋯
L(s)  = 1  + (−0.789 + 0.614i)2-s − 0.577i·3-s + (0.245 − 0.969i)4-s − 1.09·5-s + (0.354 + 0.455i)6-s − 1.74i·7-s + (0.401 + 0.915i)8-s − 0.333·9-s + (0.866 − 0.674i)10-s − 1.86·11-s + (−0.559 − 0.141i)12-s + (−0.175 − 0.984i)13-s + (1.07 + 1.38i)14-s + 0.634i·15-s + (−0.879 − 0.476i)16-s + 1.22·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.555 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $-0.555 - 0.831i$
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: \((2,\ 312,\ (\ :3/2),\ -0.555 - 0.831i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.06574321469\)
\(L(\frac12)\) \(\approx\) \(0.06574321469\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (2.23 - 1.73i)T \)
3 \( 1 + 3iT \)
13 \( 1 + (8.22 + 46.1i)T \)
good5 \( 1 + 12.2T + 125T^{2} \)
7 \( 1 + 32.4iT - 343T^{2} \)
11 \( 1 + 68.1T + 1.33e3T^{2} \)
17 \( 1 - 86.0T + 4.91e3T^{2} \)
19 \( 1 - 16.5T + 6.85e3T^{2} \)
23 \( 1 + 41.2T + 1.21e4T^{2} \)
29 \( 1 + 21.6iT - 2.43e4T^{2} \)
31 \( 1 - 111. iT - 2.97e4T^{2} \)
37 \( 1 - 410.T + 5.06e4T^{2} \)
41 \( 1 + 253. iT - 6.89e4T^{2} \)
43 \( 1 - 341. iT - 7.95e4T^{2} \)
47 \( 1 - 360. iT - 1.03e5T^{2} \)
53 \( 1 + 226. iT - 1.48e5T^{2} \)
59 \( 1 + 725.T + 2.05e5T^{2} \)
61 \( 1 - 118. iT - 2.26e5T^{2} \)
67 \( 1 + 685.T + 3.00e5T^{2} \)
71 \( 1 + 312. iT - 3.57e5T^{2} \)
73 \( 1 - 736. iT - 3.89e5T^{2} \)
79 \( 1 + 354.T + 4.93e5T^{2} \)
83 \( 1 - 798.T + 5.71e5T^{2} \)
89 \( 1 - 509. iT - 7.04e5T^{2} \)
97 \( 1 + 1.79e3iT - 9.12e5T^{2} \)
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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

−10.54311927386405264345108312834, −9.865953505906182825041657121590, −8.011091639139828668724218196519, −7.81461889917859228706071153137, −7.29775127926620531975387626796, −5.84736473493118618288594655199, −4.64303544426031398575138067658, −3.05913936422437907915696866197, −0.946556906267195167939346255204, −0.03995882831199853769627765786, 2.31755955365996856523048871732, 3.24931916640187728568383820251, 4.64274634581402692862125343988, 5.84871008934797164018010797847, 7.63859502248739759243181344168, 8.138551050358704304036879985217, 9.124398264603493330820534900502, 9.926985021613636225027102808723, 10.95743604216601079222430625217, 11.86528001664524334920214749340

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