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Properties

Label 2-462-33.32-c1-0-1
Degree $2$
Conductor $462$
Sign $-0.593 - 0.805i$
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 + (0.482 + 1.66i)3-s + 4-s + 0.885i·5-s + (−0.482 − 1.66i)6-s − i·7-s − 8-s + (−2.53 + 1.60i)9-s − 0.885i·10-s + (1.14 + 3.11i)11-s + (0.482 + 1.66i)12-s + 1.32i·13-s + i·14-s + (−1.47 + 0.426i)15-s + 16-s − 1.35·17-s + ⋯
L(s)  = 1  − 0.707·2-s + (0.278 + 0.960i)3-s + 0.5·4-s + 0.395i·5-s + (−0.196 − 0.679i)6-s − 0.377i·7-s − 0.353·8-s + (−0.845 + 0.534i)9-s − 0.279i·10-s + (0.345 + 0.938i)11-s + (0.139 + 0.480i)12-s + 0.368i·13-s + 0.267i·14-s + (−0.380 + 0.110i)15-s + 0.250·16-s − 0.329·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.593 - 0.805i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.593 - 0.805i$
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: \((2,\ 462,\ (\ :1/2),\ -0.593 - 0.805i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420024 + 0.830990i\)
\(L(\frac12)\) \(\approx\) \(0.420024 + 0.830990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.482 - 1.66i)T \)
7 \( 1 + iT \)
11 \( 1 + (-1.14 - 3.11i)T \)
good5 \( 1 - 0.885iT - 5T^{2} \)
13 \( 1 - 1.32iT - 13T^{2} \)
17 \( 1 + 1.35T + 17T^{2} \)
19 \( 1 - 4.67iT - 19T^{2} \)
23 \( 1 - 1.08iT - 23T^{2} \)
29 \( 1 + 7.92T + 29T^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 - 0.592T + 37T^{2} \)
41 \( 1 - 1.73T + 41T^{2} \)
43 \( 1 - 0.991iT - 43T^{2} \)
47 \( 1 - 0.204iT - 47T^{2} \)
53 \( 1 - 1.07iT - 53T^{2} \)
59 \( 1 - 5.25iT - 59T^{2} \)
61 \( 1 - 11.6iT - 61T^{2} \)
67 \( 1 - 12.6T + 67T^{2} \)
71 \( 1 + 14.0iT - 71T^{2} \)
73 \( 1 - 6.28iT - 73T^{2} \)
79 \( 1 + 5.69iT - 79T^{2} \)
83 \( 1 - 9.95T + 83T^{2} \)
89 \( 1 + 6.72iT - 89T^{2} \)
97 \( 1 + 3.90T + 97T^{2} \)
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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

−11.03194041327491720629177071476, −10.36002252595266070883291576065, −9.582086614233734539643811766005, −8.928408324437715649332789715020, −7.80170388854085740625336056560, −6.98542676794166871250947527857, −5.76581275761388937752616746592, −4.44636769518570400341956210641, −3.44931105707305201090059570924, −1.99907504728769682787969289511, 0.69245159263392142834254933445, 2.18604905905274320691309441904, 3.40300511044889341928815479493, 5.27991953916350640403676718743, 6.29698684847699958566126836870, 7.15842460192527492733608192299, 8.175926519190337480224346227005, 8.835421699087416991626632638180, 9.463311387949514137331940993913, 11.00029095967030828323809424735

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