Properties

Label 2-287-41.40-c1-0-5
Degree $2$
Conductor $287$
Sign $-0.478 - 0.878i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
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  + 0.889·2-s + 2.45i·3-s − 1.20·4-s + 0.645·5-s + 2.18i·6-s + i·7-s − 2.85·8-s − 3.01·9-s + 0.574·10-s + 3.84i·11-s − 2.96i·12-s + 2.91i·13-s + 0.889i·14-s + 1.58i·15-s − 0.121·16-s − 3.75i·17-s + ⋯
L(s)  = 1  + 0.628·2-s + 1.41i·3-s − 0.604·4-s + 0.288·5-s + 0.890i·6-s + 0.377i·7-s − 1.00·8-s − 1.00·9-s + 0.181·10-s + 1.15i·11-s − 0.855i·12-s + 0.808i·13-s + 0.237i·14-s + 0.408i·15-s − 0.0303·16-s − 0.910i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.478 - 0.878i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.478 - 0.878i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.478 - 0.878i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714168 + 1.20257i\)
\(L(\frac12)\) \(\approx\) \(0.714168 + 1.20257i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
41 \( 1 + (-5.62 + 3.06i)T \)
good2 \( 1 - 0.889T + 2T^{2} \)
3 \( 1 - 2.45iT - 3T^{2} \)
5 \( 1 - 0.645T + 5T^{2} \)
11 \( 1 - 3.84iT - 11T^{2} \)
13 \( 1 - 2.91iT - 13T^{2} \)
17 \( 1 + 3.75iT - 17T^{2} \)
19 \( 1 + 3.44iT - 19T^{2} \)
23 \( 1 - 3.35T + 23T^{2} \)
29 \( 1 - 5.39iT - 29T^{2} \)
31 \( 1 - 6.35T + 31T^{2} \)
37 \( 1 - 5.14T + 37T^{2} \)
43 \( 1 - 10.5T + 43T^{2} \)
47 \( 1 + 6.65iT - 47T^{2} \)
53 \( 1 + 2.47iT - 53T^{2} \)
59 \( 1 - 6.30T + 59T^{2} \)
61 \( 1 + 11.1T + 61T^{2} \)
67 \( 1 - 3.57iT - 67T^{2} \)
71 \( 1 - 8.70iT - 71T^{2} \)
73 \( 1 + 6.04T + 73T^{2} \)
79 \( 1 + 6.53iT - 79T^{2} \)
83 \( 1 + 13.3T + 83T^{2} \)
89 \( 1 - 1.08iT - 89T^{2} \)
97 \( 1 + 13.8iT - 97T^{2} \)
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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

−12.12871113110360457246668120350, −11.22021886984649448681548999435, −9.983113368458188204451202352737, −9.405246124855350367424230374312, −8.819697435828654529296147840156, −7.10509566192074857787278583624, −5.66920129256487186037793726400, −4.74170130107980534531921607807, −4.20313398631127862755777917570, −2.76750286080905500384408081578, 0.940100185991800980986837497682, 2.83435754709374218739923272395, 4.17991434253598939917359518330, 5.88019796666282969913253317687, 6.12799405611511581328692727752, 7.75546884918820654930438914003, 8.279467526067505198538436368563, 9.547091189489784902940441914548, 10.75754711073382869319642554421, 11.89901171148845996605697053914

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