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Properties

Label 2-287-287.286-c2-0-46
Degree $2$
Conductor $287$
Sign $0.949 + 0.312i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.30·2-s + 2.47·3-s + 6.90·4-s − 6.64i·5-s + 8.16·6-s + (3.90 + 5.80i)7-s + 9.60·8-s − 2.88·9-s − 21.9i·10-s + 5.07i·11-s + 17.0·12-s − 5.80·13-s + (12.9 + 19.1i)14-s − 16.4i·15-s + 4.08·16-s − 0.857·17-s + ⋯
L(s)  = 1  + 1.65·2-s + 0.823·3-s + 1.72·4-s − 1.32i·5-s + 1.36·6-s + (0.558 + 0.829i)7-s + 1.20·8-s − 0.321·9-s − 2.19i·10-s + 0.461i·11-s + 1.42·12-s − 0.446·13-s + (0.922 + 1.36i)14-s − 1.09i·15-s + 0.255·16-s − 0.0504·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.949 + 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.949 + 0.312i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ 0.949 + 0.312i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(4.85598 - 0.779118i\)
\(L(\frac12)\) \(\approx\) \(4.85598 - 0.779118i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-3.90 - 5.80i)T \)
41 \( 1 + (32.3 - 25.1i)T \)
good2 \( 1 - 3.30T + 4T^{2} \)
3 \( 1 - 2.47T + 9T^{2} \)
5 \( 1 + 6.64iT - 25T^{2} \)
11 \( 1 - 5.07iT - 121T^{2} \)
13 \( 1 + 5.80T + 169T^{2} \)
17 \( 1 + 0.857T + 289T^{2} \)
19 \( 1 + 9.47T + 361T^{2} \)
23 \( 1 - 26.2T + 529T^{2} \)
29 \( 1 - 6.33iT - 841T^{2} \)
31 \( 1 - 1.53iT - 961T^{2} \)
37 \( 1 - 46.1T + 1.36e3T^{2} \)
43 \( 1 + 24.1T + 1.84e3T^{2} \)
47 \( 1 - 18.6T + 2.20e3T^{2} \)
53 \( 1 - 77.1iT - 2.80e3T^{2} \)
59 \( 1 - 20.4iT - 3.48e3T^{2} \)
61 \( 1 + 90.4iT - 3.72e3T^{2} \)
67 \( 1 + 103. iT - 4.48e3T^{2} \)
71 \( 1 + 68.8iT - 5.04e3T^{2} \)
73 \( 1 + 47.8iT - 5.32e3T^{2} \)
79 \( 1 + 105. iT - 6.24e3T^{2} \)
83 \( 1 - 43.6iT - 6.88e3T^{2} \)
89 \( 1 + 75.9T + 7.92e3T^{2} \)
97 \( 1 - 152.T + 9.40e3T^{2} \)
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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

−12.01773944733418884952684461540, −11.10137079528802318254612358099, −9.346286326609718403763232707455, −8.699690492423803131533309922558, −7.67083107301669408826793580094, −6.15693661581026619256967517632, −5.07773060632412056985054171418, −4.56706176405909664274821022916, −3.09890954344464607079053590960, −1.97284179235906239337212882365, 2.40334434303264168015258949309, 3.22303262460616376564672283702, 4.15623216552707901272583330882, 5.46095847149395898590402490959, 6.66858503165314069002394903468, 7.36290387148823360134996187860, 8.556329180050110559410751704061, 10.09780993538477062090406740776, 11.13743572198958890171269565955, 11.52601803399614094051767657546

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