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Properties

Label 2-462-231.32-c1-0-19
Degree $2$
Conductor $462$
Sign $0.172 + 0.984i$
Analytic cond. $3.68908$
Root an. cond. $1.92070$
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  + (−0.5 − 0.866i)2-s + (1.43 − 0.972i)3-s + (−0.499 + 0.866i)4-s + (−3.72 + 2.15i)5-s + (−1.55 − 0.755i)6-s + (2.62 − 0.301i)7-s + 0.999·8-s + (1.11 − 2.78i)9-s + (3.72 + 2.15i)10-s + (2.78 − 1.80i)11-s + (0.125 + 1.72i)12-s − 2.22i·13-s + (−1.57 − 2.12i)14-s + (−3.24 + 6.70i)15-s + (−0.5 − 0.866i)16-s + (2.17 − 3.76i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.827 − 0.561i)3-s + (−0.249 + 0.433i)4-s + (−1.66 + 0.961i)5-s + (−0.636 − 0.308i)6-s + (0.993 − 0.113i)7-s + 0.353·8-s + (0.370 − 0.929i)9-s + (1.17 + 0.679i)10-s + (0.839 − 0.543i)11-s + (0.0361 + 0.498i)12-s − 0.617i·13-s + (−0.420 − 0.568i)14-s + (−0.838 + 1.73i)15-s + (−0.125 − 0.216i)16-s + (0.527 − 0.914i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.172 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.172 + 0.984i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (263, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.172 + 0.984i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00683 - 0.845665i\)
\(L(\frac12)\) \(\approx\) \(1.00683 - 0.845665i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (-1.43 + 0.972i)T \)
7 \( 1 + (-2.62 + 0.301i)T \)
11 \( 1 + (-2.78 + 1.80i)T \)
good5 \( 1 + (3.72 - 2.15i)T + (2.5 - 4.33i)T^{2} \)
13 \( 1 + 2.22iT - 13T^{2} \)
17 \( 1 + (-2.17 + 3.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.35 + 0.779i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.48 + 0.859i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 4.52T + 29T^{2} \)
31 \( 1 + (1.83 - 3.17i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.36 - 5.83i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 10.0T + 41T^{2} \)
43 \( 1 - 4.66iT - 43T^{2} \)
47 \( 1 + (3.24 - 1.87i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.14 - 1.23i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.19 - 1.84i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (-4.02 + 2.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.08 - 3.61i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 7.07iT - 71T^{2} \)
73 \( 1 + (12.7 + 7.33i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (8.48 - 4.89i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 - 17.6T + 83T^{2} \)
89 \( 1 + (10.8 - 6.24i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + 0.541T + 97T^{2} \)
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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

−11.05479991418009054803409930821, −10.07473698577233291886634319573, −8.754640589203359138447055087692, −8.142578696153408538256409162698, −7.47901638684124573665652258098, −6.69779681613630690596363918814, −4.64031518064113330954892133181, −3.50253334964755192199613224434, −2.87301688715916454001628355192, −1.00737659352335137322276244178, 1.52174436769428661749249049500, 3.78169866488663538672253486153, 4.38219680760485336263624552544, 5.26506308200647567126685780459, 7.11588611833623341499304266053, 7.82907891104704061244285669814, 8.542483355199491776142828564748, 9.005033075985100892613794904661, 10.14013137809494762073074468502, 11.32658312388379189549386514194

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