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Properties

Label 2-287-7.4-c1-0-23
Degree $2$
Conductor $287$
Sign $-0.890 + 0.455i$
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  + (−0.557 − 0.965i)2-s + (1.37 − 2.38i)3-s + (0.377 − 0.654i)4-s + (0.672 + 1.16i)5-s − 3.07·6-s + (−2.64 + 0.0328i)7-s − 3.07·8-s + (−2.29 − 3.97i)9-s + (0.749 − 1.29i)10-s + (0.538 − 0.932i)11-s + (−1.04 − 1.80i)12-s − 0.966·13-s + (1.50 + 2.53i)14-s + 3.70·15-s + (0.958 + 1.66i)16-s + (3.26 − 5.65i)17-s + ⋯
L(s)  = 1  + (−0.394 − 0.683i)2-s + (0.795 − 1.37i)3-s + (0.188 − 0.327i)4-s + (0.300 + 0.520i)5-s − 1.25·6-s + (−0.999 + 0.0124i)7-s − 1.08·8-s + (−0.764 − 1.32i)9-s + (0.237 − 0.410i)10-s + (0.162 − 0.281i)11-s + (−0.300 − 0.520i)12-s − 0.267·13-s + (0.402 + 0.678i)14-s + 0.956·15-s + (0.239 + 0.415i)16-s + (0.791 − 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.890 + 0.455i$
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: \((2,\ 287,\ (\ :1/2),\ -0.890 + 0.455i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.303534 - 1.26038i\)
\(L(\frac12)\) \(\approx\) \(0.303534 - 1.26038i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (2.64 - 0.0328i)T \)
41 \( 1 - T \)
good2 \( 1 + (0.557 + 0.965i)T + (-1 + 1.73i)T^{2} \)
3 \( 1 + (-1.37 + 2.38i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.672 - 1.16i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.538 + 0.932i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 0.966T + 13T^{2} \)
17 \( 1 + (-3.26 + 5.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.53 - 6.13i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.449 + 0.777i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 - 2.82T + 29T^{2} \)
31 \( 1 + (-0.694 + 1.20i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-3.56 - 6.17i)T + (-18.5 + 32.0i)T^{2} \)
43 \( 1 - 1.23T + 43T^{2} \)
47 \( 1 + (3.92 + 6.80i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.29 - 5.70i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (1.36 - 2.35i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.52 - 9.56i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.22 - 2.12i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 2.40T + 71T^{2} \)
73 \( 1 + (4.06 - 7.03i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.980 - 1.69i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 9.30T + 83T^{2} \)
89 \( 1 + (6.39 + 11.0i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 18.4T + 97T^{2} \)
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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

−11.72053160516398699994134780075, −10.19659879420609577558607065138, −9.723357732599088362423823363381, −8.637619996709020688398209253324, −7.47069355667360815516001735099, −6.62150714419549855635532645817, −5.80073965262036230003091821290, −3.18001296470087586605468346147, −2.56990587853739796940123717238, −1.07342196847956152379953774635, 2.86412220769562292435633786984, 3.76931508871353571398001894317, 5.12894583629694446654679221247, 6.36490868336561193196616649554, 7.58236007059666485456695762929, 8.619857452169837023732618804742, 9.363164867148766793081025994768, 9.809155317576116198297192575879, 11.00621937615998628363371647187, 12.36568377254024909696925869349

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