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Properties

Label 2-312-312.155-c1-0-18
Degree $2$
Conductor $312$
Sign $0.113 - 0.993i$
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  + (−0.752 + 1.19i)2-s + 1.73·3-s + (−0.866 − 1.80i)4-s + 1.75i·5-s + (−1.30 + 2.07i)6-s + (2.81 + 0.320i)8-s + 2.99·9-s + (−2.09 − 1.31i)10-s + 1.10·11-s + (−1.49 − 3.12i)12-s + 3.60i·13-s + 3.03i·15-s + (−2.49 + 3.12i)16-s + (−2.25 + 3.59i)18-s + (3.15 − 1.51i)20-s + ⋯
L(s)  = 1  + (−0.532 + 0.846i)2-s + 1.00·3-s + (−0.433 − 0.901i)4-s + 0.783i·5-s + (−0.532 + 0.846i)6-s + (0.993 + 0.113i)8-s + 0.999·9-s + (−0.663 − 0.417i)10-s + 0.332·11-s + (−0.433 − 0.901i)12-s + 0.999i·13-s + 0.783i·15-s + (−0.624 + 0.780i)16-s + (−0.532 + 0.846i)18-s + (0.706 − 0.339i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.113 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.113 - 0.993i$
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: \((2,\ 312,\ (\ :1/2),\ 0.113 - 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02255 + 0.912477i\)
\(L(\frac12)\) \(\approx\) \(1.02255 + 0.912477i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.752 - 1.19i)T \)
3 \( 1 - 1.73T \)
13 \( 1 - 3.60iT \)
good5 \( 1 - 1.75iT - 5T^{2} \)
7 \( 1 + 7T^{2} \)
11 \( 1 - 1.10T + 11T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + 10.0iT - 47T^{2} \)
53 \( 1 + 53T^{2} \)
59 \( 1 - 15.3T + 59T^{2} \)
61 \( 1 + 7.21iT - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 16.1iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 14.4iT - 79T^{2} \)
83 \( 1 - 13.1T + 83T^{2} \)
89 \( 1 + 18.3T + 89T^{2} \)
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

−11.76150149770154442512910280389, −10.59076256896335526656347082120, −9.790944930264312309751835938411, −8.945057030592704456380734094954, −8.134240031510238869984890398152, −7.01871311485049090742622347547, −6.54973789814329705904702681845, −4.88138888580028716771048268666, −3.57600495375720843995593337361, −1.90734758367735076617752238152, 1.27820662276913007429211099988, 2.78094096438367197950775498920, 3.90178490701435354166451066194, 5.06172305045338605043609454373, 7.00001323758669918081997652120, 8.159472256387745527276538829101, 8.610871585631877856682433919312, 9.597446181139234240452748309724, 10.27432909152856690506540220694, 11.44252687879254138984884601749

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