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Properties

Label 2-312-312.155-c1-0-30
Degree $2$
Conductor $312$
Sign $0.443 + 0.896i$
Analytic cond. $2.49133$
Root an. cond. $1.57839$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s + (−1.55 + 0.767i)3-s − 2.00·4-s − 0.380i·5-s + (−1.08 − 2.19i)6-s − 2.28·7-s − 2.82i·8-s + (1.82 − 2.38i)9-s + 0.538·10-s + (3.10 − 1.53i)12-s − 3.60·13-s − 3.23i·14-s + (0.292 + 0.591i)15-s + 4.00·16-s − 7.83i·17-s + (3.37 + 2.57i)18-s + ⋯
L(s)  = 1  + 0.999i·2-s + (−0.896 + 0.443i)3-s − 1.00·4-s − 0.170i·5-s + (−0.443 − 0.896i)6-s − 0.863·7-s − 1.00i·8-s + (0.607 − 0.794i)9-s + 0.170·10-s + (0.896 − 0.443i)12-s − 1.00·13-s − 0.863i·14-s + (0.0754 + 0.152i)15-s + 1.00·16-s − 1.90i·17-s + (0.794 + 0.607i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.443 + 0.896i$
Analytic conductor: \(2.49133\)
Root analytic conductor: \(1.57839\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :1/2),\ 0.443 + 0.896i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.219701 - 0.136482i\)
\(L(\frac12)\) \(\approx\) \(0.219701 - 0.136482i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
3 \( 1 + (1.55 - 0.767i)T \)
13 \( 1 + 3.60T \)
good5 \( 1 + 0.380iT - 5T^{2} \)
7 \( 1 + 2.28T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
17 \( 1 + 7.83iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 7.21T + 31T^{2} \)
37 \( 1 + 8.79T + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 10.3T + 43T^{2} \)
47 \( 1 + 13.5iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 12.7iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−11.65133541419484359875122874800, −10.25554400946308791614960573156, −9.623345201683286626388382891229, −8.818750415132438868678351804107, −7.16899783264523922554126518871, −6.79526842487234506650208873690, −5.40694147452888918360582136569, −4.86630223078032223238333588576, −3.42488526538839223362232812270, −0.20697619039409394745641629213, 1.78968380181422057476564394585, 3.35808257439132874705872679781, 4.70490995987245699692893181151, 5.82516923682762429920774963099, 6.88107832735138830933501735548, 8.111510804251233655252644826224, 9.347540266538928316820560224185, 10.34418058825359334254738814091, 10.81103081084552777747365829438, 11.94574708215901236145182258630

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