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Properties

Label 2-287-41.9-c1-0-8
Degree $2$
Conductor $287$
Sign $0.951 - 0.306i$
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  − 1.13i·2-s + (1.99 + 1.99i)3-s + 0.707·4-s + 2.87i·5-s + (2.27 − 2.27i)6-s + (−0.707 − 0.707i)7-s − 3.07i·8-s + 4.97i·9-s + 3.26·10-s + (−4.40 − 4.40i)11-s + (1.41 + 1.41i)12-s + (2.22 + 2.22i)13-s + (−0.804 + 0.804i)14-s + (−5.73 + 5.73i)15-s − 2.08·16-s + (2.89 − 2.89i)17-s + ⋯
L(s)  = 1  − 0.804i·2-s + (1.15 + 1.15i)3-s + 0.353·4-s + 1.28i·5-s + (0.926 − 0.926i)6-s + (−0.267 − 0.267i)7-s − 1.08i·8-s + 1.65i·9-s + 1.03·10-s + (−1.32 − 1.32i)11-s + (0.407 + 0.407i)12-s + (0.615 + 0.615i)13-s + (−0.214 + 0.214i)14-s + (−1.48 + 1.48i)15-s − 0.521·16-s + (0.701 − 0.701i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 - 0.306i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.951 - 0.306i$
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: \((2,\ 287,\ (\ :1/2),\ 0.951 - 0.306i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.91817 + 0.300819i\)
\(L(\frac12)\) \(\approx\) \(1.91817 + 0.300819i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.64 + 4.40i)T \)
good2 \( 1 + 1.13iT - 2T^{2} \)
3 \( 1 + (-1.99 - 1.99i)T + 3iT^{2} \)
5 \( 1 - 2.87iT - 5T^{2} \)
11 \( 1 + (4.40 + 4.40i)T + 11iT^{2} \)
13 \( 1 + (-2.22 - 2.22i)T + 13iT^{2} \)
17 \( 1 + (-2.89 + 2.89i)T - 17iT^{2} \)
19 \( 1 + (3.38 - 3.38i)T - 19iT^{2} \)
23 \( 1 - 3.51T + 23T^{2} \)
29 \( 1 + (2.53 + 2.53i)T + 29iT^{2} \)
31 \( 1 + 6.20T + 31T^{2} \)
37 \( 1 - 6.71T + 37T^{2} \)
43 \( 1 + 6.80iT - 43T^{2} \)
47 \( 1 + (4.76 - 4.76i)T - 47iT^{2} \)
53 \( 1 + (2.69 + 2.69i)T + 53iT^{2} \)
59 \( 1 - 2.58T + 59T^{2} \)
61 \( 1 + 2.13iT - 61T^{2} \)
67 \( 1 + (8.23 - 8.23i)T - 67iT^{2} \)
71 \( 1 + (0.550 + 0.550i)T + 71iT^{2} \)
73 \( 1 - 6.65iT - 73T^{2} \)
79 \( 1 + (-7.73 - 7.73i)T + 79iT^{2} \)
83 \( 1 + 15.0T + 83T^{2} \)
89 \( 1 + (-0.942 - 0.942i)T + 89iT^{2} \)
97 \( 1 + (13.1 - 13.1i)T - 97iT^{2} \)
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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

−11.26043996118155112581812194842, −10.84787644405517107033813382208, −10.20586990283291054209915574478, −9.366570916296728216196041557587, −8.177458758392260500854079024889, −7.15313040482023566801568793583, −5.82186200222371674908840359853, −3.92323959765258206308895147807, −3.20282653600941947249224365036, −2.50706102534112168067077284904, 1.68383543442532090115113224182, 2.85959880461260090550453723309, 4.87204514308418678678699629273, 5.97841937009258443512917837300, 7.19327987781868610680244911797, 7.88511144353396484895818988134, 8.486720853179118816969219326184, 9.393891678451559570206509636603, 10.84891521723174249928961727437, 12.33957809859427787289199007585

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