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Properties

Label 2-287-41.32-c1-0-13
Degree $2$
Conductor $287$
Sign $-0.774 + 0.632i$
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  − 1.70i·2-s + (−1.24 + 1.24i)3-s − 0.907·4-s − 2.38i·5-s + (2.11 + 2.11i)6-s + (0.707 − 0.707i)7-s − 1.86i·8-s − 0.0865i·9-s − 4.05·10-s + (−1.55 + 1.55i)11-s + (1.12 − 1.12i)12-s + (4.13 − 4.13i)13-s + (−1.20 − 1.20i)14-s + (2.95 + 2.95i)15-s − 4.99·16-s + (−5.23 − 5.23i)17-s + ⋯
L(s)  = 1  − 1.20i·2-s + (−0.717 + 0.717i)3-s − 0.453·4-s − 1.06i·5-s + (0.864 + 0.864i)6-s + (0.267 − 0.267i)7-s − 0.658i·8-s − 0.0288i·9-s − 1.28·10-s + (−0.467 + 0.467i)11-s + (0.325 − 0.325i)12-s + (1.14 − 1.14i)13-s + (−0.322 − 0.322i)14-s + (0.763 + 0.763i)15-s − 1.24·16-s + (−1.26 − 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 + 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.774 + 0.632i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.774 + 0.632i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.328718 - 0.922056i\)
\(L(\frac12)\) \(\approx\) \(0.328718 - 0.922056i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.707 + 0.707i)T \)
41 \( 1 + (-6.22 + 1.50i)T \)
good2 \( 1 + 1.70iT - 2T^{2} \)
3 \( 1 + (1.24 - 1.24i)T - 3iT^{2} \)
5 \( 1 + 2.38iT - 5T^{2} \)
11 \( 1 + (1.55 - 1.55i)T - 11iT^{2} \)
13 \( 1 + (-4.13 + 4.13i)T - 13iT^{2} \)
17 \( 1 + (5.23 + 5.23i)T + 17iT^{2} \)
19 \( 1 + (2.42 + 2.42i)T + 19iT^{2} \)
23 \( 1 + 1.83T + 23T^{2} \)
29 \( 1 + (2.44 - 2.44i)T - 29iT^{2} \)
31 \( 1 - 9.33T + 31T^{2} \)
37 \( 1 - 7.47T + 37T^{2} \)
43 \( 1 - 10.8iT - 43T^{2} \)
47 \( 1 + (-7.24 - 7.24i)T + 47iT^{2} \)
53 \( 1 + (-3.23 + 3.23i)T - 53iT^{2} \)
59 \( 1 + 8.54T + 59T^{2} \)
61 \( 1 - 2.64iT - 61T^{2} \)
67 \( 1 + (3.65 + 3.65i)T + 67iT^{2} \)
71 \( 1 + (1.73 - 1.73i)T - 71iT^{2} \)
73 \( 1 - 1.97iT - 73T^{2} \)
79 \( 1 + (-7.94 + 7.94i)T - 79iT^{2} \)
83 \( 1 - 16.2T + 83T^{2} \)
89 \( 1 + (8.80 - 8.80i)T - 89iT^{2} \)
97 \( 1 + (5.70 + 5.70i)T + 97iT^{2} \)
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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

−11.10145523918729431589994474779, −10.89418794338760092810512481516, −9.855544732684965619333525961614, −8.987590807640486309203673924721, −7.78444835979191215466085259152, −6.18574498361458540425767831324, −4.82614399133828061511319596124, −4.33881701545717436194749642831, −2.62468491158194987867375781155, −0.814605654402133249126787450086, 2.15497745027244642715126318122, 4.14107550231855025919492549738, 5.95562959439480603994609889452, 6.23166189530923242384172923229, 6.97506402471718757048371908723, 8.080546748367910733463018250071, 8.900008073476098215064665618006, 10.68764793412319458677021396034, 11.18891634514282995698426396494, 12.05625752949176580779391455575

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