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Properties

Label 2-287-7.4-c1-0-3
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: $\chi_{287} (165, \cdot )$
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

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

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