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Properties

Label 2-287-41.10-c1-0-6
Degree $2$
Conductor $287$
Sign $0.938 + 0.344i$
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  + (−1.52 − 1.10i)2-s + 2.82·3-s + (0.477 + 1.46i)4-s + (0.756 + 2.32i)5-s + (−4.30 − 3.12i)6-s + (−0.809 + 0.587i)7-s + (−0.264 + 0.814i)8-s + 4.97·9-s + (1.42 − 4.38i)10-s + (0.847 − 2.60i)11-s + (1.34 + 4.15i)12-s + (2.49 + 1.81i)13-s + 1.88·14-s + (2.13 + 6.57i)15-s + (3.80 − 2.76i)16-s + (−0.693 + 2.13i)17-s + ⋯
L(s)  = 1  + (−1.07 − 0.782i)2-s + 1.63·3-s + (0.238 + 0.734i)4-s + (0.338 + 1.04i)5-s + (−1.75 − 1.27i)6-s + (−0.305 + 0.222i)7-s + (−0.0935 + 0.287i)8-s + 1.65·9-s + (0.450 − 1.38i)10-s + (0.255 − 0.786i)11-s + (0.389 + 1.19i)12-s + (0.692 + 0.503i)13-s + 0.503·14-s + (0.551 + 1.69i)15-s + (0.951 − 0.691i)16-s + (−0.168 + 0.517i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 + 0.344i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.938 + 0.344i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.938 + 0.344i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27319 - 0.226210i\)
\(L(\frac12)\) \(\approx\) \(1.27319 - 0.226210i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (3.39 + 5.42i)T \)
good2 \( 1 + (1.52 + 1.10i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 - 2.82T + 3T^{2} \)
5 \( 1 + (-0.756 - 2.32i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (-0.847 + 2.60i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-2.49 - 1.81i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (0.693 - 2.13i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (0.529 - 0.384i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (2.53 + 1.84i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-2.57 - 7.93i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-2.48 + 7.64i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.28 + 7.04i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (5.22 + 3.79i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (1.92 + 1.39i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-3.32 - 10.2i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (12.1 + 8.82i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-5.94 + 4.32i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (-1.15 - 3.54i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (4.53 - 13.9i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + 7.45T + 73T^{2} \)
79 \( 1 + 5.60T + 79T^{2} \)
83 \( 1 + 6.92T + 83T^{2} \)
89 \( 1 + (-2.83 + 2.06i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-1.44 - 4.45i)T + (-78.4 + 57.0i)T^{2} \)
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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

−11.37584372846771245879037809618, −10.52345847514671204449588252640, −9.844239092970828763855163801993, −8.772249501333282762598097205666, −8.554864964134645025145970478157, −7.27110878449478879419560436651, −6.08339243936930832024209231440, −3.72598848830393866387259909209, −2.83701682170619318956215254437, −1.85934506279740889550276416636, 1.44872606599230738388508146826, 3.25231991444283372656410449859, 4.61299715062220411602008941511, 6.36303440676266150978673985682, 7.42767262455174572865534063123, 8.322643374127777194858129228861, 8.754222326583389021439461019291, 9.674828641134999241569214566701, 10.09478823678974404314540046335, 12.07341123934345523029209344706

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