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Properties

Label 2-462-77.76-c1-0-10
Degree $2$
Conductor $462$
Sign $0.499 + 0.866i$
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 + 0.266i·5-s + 6-s + (1.34 − 2.27i)7-s + i·8-s − 9-s + 0.266·10-s + (1.62 − 2.89i)11-s − i·12-s − 2.55·13-s + (−2.27 − 1.34i)14-s − 0.266·15-s + 16-s + 4.96·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 0.118i·5-s + 0.408·6-s + (0.509 − 0.860i)7-s + 0.353i·8-s − 0.333·9-s + 0.0841·10-s + (0.489 − 0.871i)11-s − 0.288i·12-s − 0.707·13-s + (−0.608 − 0.360i)14-s − 0.0687·15-s + 0.250·16-s + 1.20·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.499 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(462\)    =    \(2 \cdot 3 \cdot 7 \cdot 11\)
Sign: $0.499 + 0.866i$
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: \((2,\ 462,\ (\ :1/2),\ 0.499 + 0.866i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.22711 - 0.708497i\)
\(L(\frac12)\) \(\approx\) \(1.22711 - 0.708497i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 + 2.27i)T \)
11 \( 1 + (-1.62 + 2.89i)T \)
good5 \( 1 - 0.266iT - 5T^{2} \)
13 \( 1 + 2.55T + 13T^{2} \)
17 \( 1 - 4.96T + 17T^{2} \)
19 \( 1 - 8.21T + 19T^{2} \)
23 \( 1 + 5.23T + 23T^{2} \)
29 \( 1 + 5.25iT - 29T^{2} \)
31 \( 1 + 4.28iT - 31T^{2} \)
37 \( 1 - 7.39T + 37T^{2} \)
41 \( 1 + 6.21T + 41T^{2} \)
43 \( 1 + 1.46iT - 43T^{2} \)
47 \( 1 - 10.0iT - 47T^{2} \)
53 \( 1 + 13.1T + 53T^{2} \)
59 \( 1 - 9.10iT - 59T^{2} \)
61 \( 1 + 5.98T + 61T^{2} \)
67 \( 1 - 12.3T + 67T^{2} \)
71 \( 1 - 4.93T + 71T^{2} \)
73 \( 1 + 2.96T + 73T^{2} \)
79 \( 1 - 6.16iT - 79T^{2} \)
83 \( 1 + 10.6T + 83T^{2} \)
89 \( 1 - 1.78iT - 89T^{2} \)
97 \( 1 - 0.146iT - 97T^{2} \)
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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

−11.00229473067806095356152545370, −9.884131974117664052384708873529, −9.600865845776238728973052867020, −8.190636382266112986943862215648, −7.51176239969090340735160249941, −5.96873273047627563744013483471, −4.92429929841770258241666688996, −3.89183355828980818903581068031, −2.94845135089928250184815367912, −1.05286590316987132590064235136, 1.53829778708774177052864469852, 3.16702872368425847591865238072, 4.87957724346189095455314989526, 5.48797860741047268713820173285, 6.69370433535108827536906188083, 7.54202686544589279040900636151, 8.247955438120417492091425455925, 9.332785777993172811759266721492, 9.977237036684081149217868594789, 11.50524905404716497181925588186

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