Properties

Label 2-287-41.9-c1-0-15
Degree $2$
Conductor $287$
Sign $-0.483 + 0.875i$
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  − 1.99i·2-s + (1.04 + 1.04i)3-s − 1.97·4-s − 1.71i·5-s + (2.08 − 2.08i)6-s + (−0.707 − 0.707i)7-s − 0.0591i·8-s − 0.801i·9-s − 3.40·10-s + (2.84 + 2.84i)11-s + (−2.06 − 2.06i)12-s + (−0.592 − 0.592i)13-s + (−1.40 + 1.40i)14-s + (1.79 − 1.79i)15-s − 4.05·16-s + (4.76 − 4.76i)17-s + ⋯
L(s)  = 1  − 1.40i·2-s + (0.605 + 0.605i)3-s − 0.985·4-s − 0.764i·5-s + (0.852 − 0.852i)6-s + (−0.267 − 0.267i)7-s − 0.0209i·8-s − 0.267i·9-s − 1.07·10-s + (0.856 + 0.856i)11-s + (−0.596 − 0.596i)12-s + (−0.164 − 0.164i)13-s + (−0.376 + 0.376i)14-s + (0.462 − 0.462i)15-s − 1.01·16-s + (1.15 − 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.483 + 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.483 + 0.875i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (50, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.483 + 0.875i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.773369 - 1.30990i\)
\(L(\frac12)\) \(\approx\) \(0.773369 - 1.30990i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.359 - 6.39i)T \)
good2 \( 1 + 1.99iT - 2T^{2} \)
3 \( 1 + (-1.04 - 1.04i)T + 3iT^{2} \)
5 \( 1 + 1.71iT - 5T^{2} \)
11 \( 1 + (-2.84 - 2.84i)T + 11iT^{2} \)
13 \( 1 + (0.592 + 0.592i)T + 13iT^{2} \)
17 \( 1 + (-4.76 + 4.76i)T - 17iT^{2} \)
19 \( 1 + (1.80 - 1.80i)T - 19iT^{2} \)
23 \( 1 + 7.37T + 23T^{2} \)
29 \( 1 + (-6.01 - 6.01i)T + 29iT^{2} \)
31 \( 1 + 7.24T + 31T^{2} \)
37 \( 1 - 8.10T + 37T^{2} \)
43 \( 1 - 10.4iT - 43T^{2} \)
47 \( 1 + (4.86 - 4.86i)T - 47iT^{2} \)
53 \( 1 + (-2.81 - 2.81i)T + 53iT^{2} \)
59 \( 1 - 1.07T + 59T^{2} \)
61 \( 1 + 3.67iT - 61T^{2} \)
67 \( 1 + (-5.59 + 5.59i)T - 67iT^{2} \)
71 \( 1 + (-8.63 - 8.63i)T + 71iT^{2} \)
73 \( 1 + 8.12iT - 73T^{2} \)
79 \( 1 + (9.11 + 9.11i)T + 79iT^{2} \)
83 \( 1 - 0.0385T + 83T^{2} \)
89 \( 1 + (-5.40 - 5.40i)T + 89iT^{2} \)
97 \( 1 + (3.41 - 3.41i)T - 97iT^{2} \)
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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

−11.66333079936714697799162431419, −10.39477341216990634798477375866, −9.553711216360355282056400921317, −9.344029156160160810172766677329, −7.993794634031850447095911045218, −6.55218186582390148805338072710, −4.76305011995321227808365447721, −3.92524086289822521056031573928, −2.89997559429701192599900524938, −1.24290263381542299964742459032, 2.29674016890426737113300574957, 3.83239207982754543715473218076, 5.60995130740814439613970423273, 6.39167752773032983600861682630, 7.20025011653110349389316559599, 8.162624734324752422636632460864, 8.695930401342294831362803930409, 10.06301959989148011920637710396, 11.20020705812729200300061024774, 12.31145410983242294101686056425

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