Properties

Label 2-462-33.32-c1-0-20
Degree $2$
Conductor $462$
Sign $0.965 + 0.261i$
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  + 2-s + (1.72 + 0.163i)3-s + 4-s − 1.55i·5-s + (1.72 + 0.163i)6-s i·7-s + 8-s + (2.94 + 0.565i)9-s − 1.55i·10-s + (−0.560 + 3.26i)11-s + (1.72 + 0.163i)12-s − 2.32i·13-s i·14-s + (0.254 − 2.68i)15-s + 16-s − 6.13·17-s + ⋯
L(s)  = 1  + 0.707·2-s + (0.995 + 0.0946i)3-s + 0.5·4-s − 0.695i·5-s + (0.703 + 0.0669i)6-s − 0.377i·7-s + 0.353·8-s + (0.982 + 0.188i)9-s − 0.491i·10-s + (−0.168 + 0.985i)11-s + (0.497 + 0.0473i)12-s − 0.645i·13-s − 0.267i·14-s + (0.0658 − 0.692i)15-s + 0.250·16-s − 1.48·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.261i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.965 + 0.261i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (197, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.965 + 0.261i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.82416 - 0.375782i\)
\(L(\frac12)\) \(\approx\) \(2.82416 - 0.375782i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + (-1.72 - 0.163i)T \)
7 \( 1 + iT \)
11 \( 1 + (0.560 - 3.26i)T \)
good5 \( 1 + 1.55iT - 5T^{2} \)
13 \( 1 + 2.32iT - 13T^{2} \)
17 \( 1 + 6.13T + 17T^{2} \)
19 \( 1 - 3.69iT - 19T^{2} \)
23 \( 1 + 4.25iT - 23T^{2} \)
29 \( 1 - 1.46T + 29T^{2} \)
31 \( 1 + 1.87T + 31T^{2} \)
37 \( 1 + 6.88T + 37T^{2} \)
41 \( 1 + 2.23T + 41T^{2} \)
43 \( 1 - 12.1iT - 43T^{2} \)
47 \( 1 - 5.48iT - 47T^{2} \)
53 \( 1 - 9.89iT - 53T^{2} \)
59 \( 1 + 6.56iT - 59T^{2} \)
61 \( 1 + 7.35iT - 61T^{2} \)
67 \( 1 + 8.79T + 67T^{2} \)
71 \( 1 + 2.88iT - 71T^{2} \)
73 \( 1 + 1.67iT - 73T^{2} \)
79 \( 1 + 12.0iT - 79T^{2} \)
83 \( 1 - 0.338T + 83T^{2} \)
89 \( 1 + 5.55iT - 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−10.90197700115210422403558015041, −10.15370895357502765478978419574, −9.148760085129283132101748847824, −8.258717556612482332290814183397, −7.39543647477026918376914719372, −6.40028507863242988479972011885, −4.83878599560099243218866008337, −4.33366881712540859763460666700, −3.01893115730309873547401585312, −1.74866350235044795566299166611, 2.09664687407650192185266966206, 3.05479973795611846248110953746, 4.02809446168922998692987682305, 5.30606830742799958889673972168, 6.67478047776068203590468141054, 7.12321883171349985433912907475, 8.523817816654045656960860796780, 9.049318663979790389679530577106, 10.34510737634187143050993707602, 11.15605986747601516646495582487

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