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Properties

Label 2-462-77.76-c1-0-6
Degree $2$
Conductor $462$
Sign $0.947 + 0.320i$
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  − i·2-s − i·3-s − 4-s + 4.14i·5-s − 6-s + (0.717 − 2.54i)7-s + i·8-s − 9-s + 4.14·10-s + (−0.170 + 3.31i)11-s + i·12-s + 3.09·13-s + (−2.54 − 0.717i)14-s + 4.14·15-s + 16-s + 3.57·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 1.85i·5-s − 0.408·6-s + (0.271 − 0.962i)7-s + 0.353i·8-s − 0.333·9-s + 1.30·10-s + (−0.0514 + 0.998i)11-s + 0.288i·12-s + 0.857·13-s + (−0.680 − 0.191i)14-s + 1.06·15-s + 0.250·16-s + 0.867·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.947 + 0.320i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.947 + 0.320i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (307, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ 0.947 + 0.320i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.39466 - 0.229400i\)
\(L(\frac12)\) \(\approx\) \(1.39466 - 0.229400i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + iT \)
7 \( 1 + (-0.717 + 2.54i)T \)
11 \( 1 + (0.170 - 3.31i)T \)
good5 \( 1 - 4.14iT - 5T^{2} \)
13 \( 1 - 3.09T + 13T^{2} \)
17 \( 1 - 3.57T + 17T^{2} \)
19 \( 1 - 3.91T + 19T^{2} \)
23 \( 1 - 7.71T + 23T^{2} \)
29 \( 1 + 1.65iT - 29T^{2} \)
31 \( 1 - 9.23iT - 31T^{2} \)
37 \( 1 + 0.869T + 37T^{2} \)
41 \( 1 + 5.91T + 41T^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 - 6.76iT - 47T^{2} \)
53 \( 1 + 1.88T + 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 + 3.37T + 61T^{2} \)
67 \( 1 - 9.84T + 67T^{2} \)
71 \( 1 - 1.33T + 71T^{2} \)
73 \( 1 + 5.57T + 73T^{2} \)
79 \( 1 - 10.8iT - 79T^{2} \)
83 \( 1 + 5.67T + 83T^{2} \)
89 \( 1 - 10.6iT - 89T^{2} \)
97 \( 1 - 4.52iT - 97T^{2} \)
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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

−10.90917838509104611011246183108, −10.42388018488529133052722424169, −9.557389316171030805160315455144, −8.093456472059848495045942728518, −7.15704647141685272986569513844, −6.74301043506926052312178489509, −5.24307857020369069055388543663, −3.68710907421986623093625690610, −2.94217768843173569315520666019, −1.47841446926573388712823224898, 1.07824718978450764219197378634, 3.37223205962518624523432784997, 4.68574924192280706062998124484, 5.42771233389563257294921703997, 5.97181316253749306745368190190, 7.79052633282349861448338170720, 8.583144372446649965083740326021, 8.997237420186397129455518178562, 9.785807300111689737236533613680, 11.27959376376638670407565027202

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