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Properties

Label 2-471-157.156-c1-0-1
Degree $2$
Conductor $471$
Sign $0.919 + 0.392i$
Analytic cond. $3.76095$
Root an. cond. $1.93931$
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  − 2.72i·2-s + 3-s − 5.40·4-s + 1.84i·5-s − 2.72i·6-s + 5.01i·7-s + 9.26i·8-s + 9-s + 5.02·10-s − 1.38·11-s − 5.40·12-s − 1.03·13-s + 13.6·14-s + 1.84i·15-s + 14.3·16-s + 5.75·17-s + ⋯
L(s)  = 1  − 1.92i·2-s + 0.577·3-s − 2.70·4-s + 0.826i·5-s − 1.11i·6-s + 1.89i·7-s + 3.27i·8-s + 0.333·9-s + 1.59·10-s − 0.416·11-s − 1.56·12-s − 0.285·13-s + 3.64·14-s + 0.477i·15-s + 3.59·16-s + 1.39·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 471 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.919 + 0.392i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(471\)    =    \(3 \cdot 157\)
Sign: $0.919 + 0.392i$
Analytic conductor: \(3.76095\)
Root analytic conductor: \(1.93931\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{471} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 471,\ (\ :1/2),\ 0.919 + 0.392i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.18591 - 0.242319i\)
\(L(\frac12)\) \(\approx\) \(1.18591 - 0.242319i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
157 \( 1 + (-11.5 - 4.91i)T \)
good2 \( 1 + 2.72iT - 2T^{2} \)
5 \( 1 - 1.84iT - 5T^{2} \)
7 \( 1 - 5.01iT - 7T^{2} \)
11 \( 1 + 1.38T + 11T^{2} \)
13 \( 1 + 1.03T + 13T^{2} \)
17 \( 1 - 5.75T + 17T^{2} \)
19 \( 1 + 5.92T + 19T^{2} \)
23 \( 1 - 2.89iT - 23T^{2} \)
29 \( 1 - 1.63iT - 29T^{2} \)
31 \( 1 + 7.31T + 31T^{2} \)
37 \( 1 - 6.34T + 37T^{2} \)
41 \( 1 - 6.98iT - 41T^{2} \)
43 \( 1 + 1.29iT - 43T^{2} \)
47 \( 1 - 6.08T + 47T^{2} \)
53 \( 1 - 5.57iT - 53T^{2} \)
59 \( 1 + 3.41iT - 59T^{2} \)
61 \( 1 + 10.6iT - 61T^{2} \)
67 \( 1 - 12.1T + 67T^{2} \)
71 \( 1 + 3.19T + 71T^{2} \)
73 \( 1 - 5.10iT - 73T^{2} \)
79 \( 1 + 7.11iT - 79T^{2} \)
83 \( 1 + 12.9iT - 83T^{2} \)
89 \( 1 + 2.52T + 89T^{2} \)
97 \( 1 + 13.9iT - 97T^{2} \)
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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

−11.02865910122970266455681424087, −10.15936172838638622337642831882, −9.388976115294298771294818375739, −8.709965808911211268689730584459, −7.80645233400276721239783597628, −5.92305938211534337348665491706, −4.92410067853641244304234119962, −3.46257820352156099678980844271, −2.72780421077548359671767708949, −1.95468128723674095545354690904, 0.73118534857296340902690699437, 3.82922255242176687414533562418, 4.46880158761889284247348270846, 5.47857416870875336280350898778, 6.72897219732108392575212774136, 7.49167082929471118075855234660, 8.059293895552010644459484478011, 8.897149268091588213565820781793, 9.868342081837263966722702241304, 10.57850828217246152298466819850

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