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Properties

Label 2-462-21.17-c1-0-5
Degree $2$
Conductor $462$
Sign $-0.848 - 0.528i$
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  + (−0.866 + 0.5i)2-s + (0.0733 + 1.73i)3-s + (0.499 − 0.866i)4-s + (1.79 + 3.10i)5-s + (−0.928 − 1.46i)6-s + (0.833 + 2.51i)7-s + 0.999i·8-s + (−2.98 + 0.253i)9-s + (−3.10 − 1.79i)10-s + (0.866 + 0.5i)11-s + (1.53 + 0.801i)12-s + 0.365i·13-s + (−1.97 − 1.75i)14-s + (−5.24 + 3.33i)15-s + (−0.5 − 0.866i)16-s + (1.04 − 1.81i)17-s + ⋯
L(s)  = 1  + (−0.612 + 0.353i)2-s + (0.0423 + 0.999i)3-s + (0.249 − 0.433i)4-s + (0.802 + 1.39i)5-s + (−0.379 − 0.596i)6-s + (0.315 + 0.949i)7-s + 0.353i·8-s + (−0.996 + 0.0846i)9-s + (−0.983 − 0.567i)10-s + (0.261 + 0.150i)11-s + (0.443 + 0.231i)12-s + 0.101i·13-s + (−0.528 − 0.469i)14-s + (−1.35 + 0.860i)15-s + (−0.125 − 0.216i)16-s + (0.254 − 0.440i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 - 0.528i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.848 - 0.528i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $-0.848 - 0.528i$
Analytic conductor: \(3.68908\)
Root analytic conductor: \(1.92070\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{462} (353, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 462,\ (\ :1/2),\ -0.848 - 0.528i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.336362 + 1.17549i\)
\(L(\frac12)\) \(\approx\) \(0.336362 + 1.17549i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 0.5i)T \)
3 \( 1 + (-0.0733 - 1.73i)T \)
7 \( 1 + (-0.833 - 2.51i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
good5 \( 1 + (-1.79 - 3.10i)T + (-2.5 + 4.33i)T^{2} \)
13 \( 1 - 0.365iT - 13T^{2} \)
17 \( 1 + (-1.04 + 1.81i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-5.84 + 3.37i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (-1.78 + 1.03i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 3.75iT - 29T^{2} \)
31 \( 1 + (8.04 + 4.64i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.32 - 9.22i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 3.58T + 41T^{2} \)
43 \( 1 - 4.30T + 43T^{2} \)
47 \( 1 + (6.39 + 11.0i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.13 + 1.80i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.85 - 6.67i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.48 + 4.32i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.98 - 5.16i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 4.43iT - 71T^{2} \)
73 \( 1 + (-4.69 - 2.70i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (3.30 + 5.72i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 7.62T + 83T^{2} \)
89 \( 1 + (-7.03 - 12.1i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 0.651iT - 97T^{2} \)
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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

−11.34051997723016318725103888990, −10.27242711709363404383385888076, −9.606493072253137460758789714672, −9.063411864800786571213957046611, −7.83571993203273177429718193556, −6.74551726839310419248125507501, −5.82981528472136735718201712265, −5.01647832170494453756901805460, −3.23567011006696649063857862414, −2.29517193137737427037287295169, 1.01204336707433453669615679974, 1.70369184049386184264485536363, 3.50112797816906050902448667687, 5.07174834689454588977368911568, 6.01771128208885744361490308130, 7.31246902956148280743838500122, 7.964630775624228593658985607171, 8.942659249220065270060864100947, 9.566377387300755542080466587578, 10.70971421154355838746999518228

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