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Properties

Label 2-287-7.2-c1-0-15
Degree $2$
Conductor $287$
Sign $0.411 + 0.911i$
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  + (−0.587 + 1.01i)2-s + (−1.36 − 2.36i)3-s + (0.309 + 0.535i)4-s + (1.41 − 2.44i)5-s + 3.20·6-s + (2.04 + 1.67i)7-s − 3.07·8-s + (−2.22 + 3.84i)9-s + (1.66 + 2.87i)10-s + (−2.35 − 4.08i)11-s + (0.843 − 1.46i)12-s + 2.95·13-s + (−2.91 + 1.09i)14-s − 7.71·15-s + (1.19 − 2.06i)16-s + (−2.53 − 4.38i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.719i)2-s + (−0.787 − 1.36i)3-s + (0.154 + 0.267i)4-s + (0.632 − 1.09i)5-s + 1.30·6-s + (0.773 + 0.634i)7-s − 1.08·8-s + (−0.740 + 1.28i)9-s + (0.525 + 0.910i)10-s + (−0.711 − 1.23i)11-s + (0.243 − 0.421i)12-s + 0.820·13-s + (−0.777 + 0.293i)14-s − 1.99·15-s + (0.297 − 0.515i)16-s + (−0.614 − 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.411 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.411 + 0.911i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (247, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.411 + 0.911i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753110 - 0.486508i\)
\(L(\frac12)\) \(\approx\) \(0.753110 - 0.486508i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.04 - 1.67i)T \)
41 \( 1 - T \)
good2 \( 1 + (0.587 - 1.01i)T + (-1 - 1.73i)T^{2} \)
3 \( 1 + (1.36 + 2.36i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-1.41 + 2.44i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 + (2.35 + 4.08i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 2.95T + 13T^{2} \)
17 \( 1 + (2.53 + 4.38i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-2.25 + 3.90i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.804 - 1.39i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 7.93T + 29T^{2} \)
31 \( 1 + (3.65 + 6.33i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (0.676 - 1.17i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 + 1.02T + 43T^{2} \)
47 \( 1 + (3.92 - 6.79i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.860 + 1.49i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-5.59 - 9.68i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.48 + 6.03i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (0.324 + 0.561i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 + (-6.04 - 10.4i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (2.72 - 4.72i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 12.1T + 83T^{2} \)
89 \( 1 + (2.96 - 5.14i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 6.86T + 97T^{2} \)
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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

−11.56292933004188102767183564941, −11.27310064725949713141727080182, −9.250265742046994879599126327697, −8.476073024339092108929138592538, −7.84398358376860637131375315792, −6.69093072831968765180804772451, −5.79512984949990039411132792447, −5.16632402947198697119698784781, −2.53454300905946144588599446933, −0.872538193927195987683354553890, 1.91413157505687429493303688683, 3.53287430318089555037816525171, 4.81524166036230247459543225969, 5.89361809377157849213074119992, 6.85694824611848030786874461300, 8.501324203694252562262548676110, 9.934089299127791761054396302875, 10.24794517365092379087447337779, 10.74705723203086651169381940124, 11.39319806346330028444039998749

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