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Properties

Label 2-462-77.76-c1-0-9
Degree $2$
Conductor $462$
Sign $0.999 - 0.0273i$
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 + 1.18i·5-s + 6-s + (1.74 − 1.99i)7-s − i·8-s − 9-s − 1.18·10-s + (−2.25 − 2.43i)11-s + i·12-s + 5.98·13-s + (1.99 + 1.74i)14-s + 1.18·15-s + 16-s + 4.29·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.577i·3-s − 0.5·4-s + 0.530i·5-s + 0.408·6-s + (0.658 − 0.752i)7-s − 0.353i·8-s − 0.333·9-s − 0.374·10-s + (−0.678 − 0.734i)11-s + 0.288i·12-s + 1.65·13-s + (0.532 + 0.465i)14-s + 0.306·15-s + 0.250·16-s + 1.04·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0273i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0273i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.999 - 0.0273i$
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: \((2,\ 462,\ (\ :1/2),\ 0.999 - 0.0273i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44979 + 0.0198598i\)
\(L(\frac12)\) \(\approx\) \(1.44979 + 0.0198598i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.74 + 1.99i)T \)
11 \( 1 + (2.25 + 2.43i)T \)
good5 \( 1 - 1.18iT - 5T^{2} \)
13 \( 1 - 5.98T + 13T^{2} \)
17 \( 1 - 4.29T + 17T^{2} \)
19 \( 1 + 0.203T + 19T^{2} \)
23 \( 1 + 3.11T + 23T^{2} \)
29 \( 1 + 2.50iT - 29T^{2} \)
31 \( 1 + 2.79iT - 31T^{2} \)
37 \( 1 - 8.96T + 37T^{2} \)
41 \( 1 - 2.20T + 41T^{2} \)
43 \( 1 - 4.37iT - 43T^{2} \)
47 \( 1 - 2.05iT - 47T^{2} \)
53 \( 1 - 1.51T + 53T^{2} \)
59 \( 1 - 7.96iT - 59T^{2} \)
61 \( 1 + 11.6T + 61T^{2} \)
67 \( 1 + 12.4T + 67T^{2} \)
71 \( 1 + 15.8T + 71T^{2} \)
73 \( 1 + 2.29T + 73T^{2} \)
79 \( 1 + 9.85iT - 79T^{2} \)
83 \( 1 - 10.0T + 83T^{2} \)
89 \( 1 - 8.87iT - 89T^{2} \)
97 \( 1 + 9.46iT - 97T^{2} \)
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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

−10.95485967137657614178909174671, −10.34562891175280478903778055984, −8.955530250363678047996601563240, −7.951064789402629153616604272449, −7.61651662891273986837866972816, −6.31594284167613288812799066773, −5.75651279352147987796689753707, −4.30227792250577007269576238407, −3.09580192147977834343612461057, −1.11290052069359868432660902566, 1.48964594292846673242596563019, 2.96684658634012975235248887970, 4.22426712016019197813510164022, 5.14605005773880659148799376651, 5.98029891835944966891410032682, 7.78293657489854765665525362203, 8.575873403722294330637112842843, 9.260190716372672492411190006178, 10.30581326831641325007177257619, 10.96599631271836121205575938646

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