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Properties

Label 2-287-41.10-c1-0-20
Degree $2$
Conductor $287$
Sign $0.973 + 0.229i$
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  + (0.991 + 0.720i)2-s + 2.24·3-s + (−0.153 − 0.473i)4-s + (−1.02 − 3.15i)5-s + (2.22 + 1.61i)6-s + (−0.809 + 0.587i)7-s + (0.946 − 2.91i)8-s + 2.03·9-s + (1.25 − 3.86i)10-s + (−1.09 + 3.37i)11-s + (−0.345 − 1.06i)12-s + (3.95 + 2.87i)13-s − 1.22·14-s + (−2.30 − 7.08i)15-s + (2.22 − 1.62i)16-s + (−1.69 + 5.20i)17-s + ⋯
L(s)  = 1  + (0.701 + 0.509i)2-s + 1.29·3-s + (−0.0769 − 0.236i)4-s + (−0.458 − 1.41i)5-s + (0.908 + 0.659i)6-s + (−0.305 + 0.222i)7-s + (0.334 − 1.02i)8-s + 0.678·9-s + (0.397 − 1.22i)10-s + (−0.330 + 1.01i)11-s + (−0.0996 − 0.306i)12-s + (1.09 + 0.796i)13-s − 0.327·14-s + (−0.594 − 1.82i)15-s + (0.557 − 0.405i)16-s + (−0.409 + 1.26i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.973 + 0.229i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.31439 - 0.268991i\)
\(L(\frac12)\) \(\approx\) \(2.31439 - 0.268991i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (4.02 + 4.98i)T \)
good2 \( 1 + (-0.991 - 0.720i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 - 2.24T + 3T^{2} \)
5 \( 1 + (1.02 + 3.15i)T + (-4.04 + 2.93i)T^{2} \)
11 \( 1 + (1.09 - 3.37i)T + (-8.89 - 6.46i)T^{2} \)
13 \( 1 + (-3.95 - 2.87i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.69 - 5.20i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.02 + 1.47i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + (0.123 + 0.0894i)T + (7.10 + 21.8i)T^{2} \)
29 \( 1 + (-0.750 - 2.30i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (1.44 - 4.44i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (2.54 + 7.81i)T + (-29.9 + 21.7i)T^{2} \)
43 \( 1 + (-3.29 - 2.39i)T + (13.2 + 40.8i)T^{2} \)
47 \( 1 + (-10.8 - 7.88i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.14 + 6.60i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (6.55 + 4.76i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-2.79 + 2.03i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + (1.73 + 5.33i)T + (-54.2 + 39.3i)T^{2} \)
71 \( 1 + (-1.64 + 5.06i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 - 1.82T + 73T^{2} \)
79 \( 1 - 8.58T + 79T^{2} \)
83 \( 1 + 15.3T + 83T^{2} \)
89 \( 1 + (14.3 - 10.4i)T + (27.5 - 84.6i)T^{2} \)
97 \( 1 + (-3.08 - 9.49i)T + (-78.4 + 57.0i)T^{2} \)
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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

−12.41893731916158697631376516725, −10.76732045592922406984048774485, −9.399819925154516834764870769730, −8.954812976257999742932229762305, −8.047854820301500197086082991318, −6.89387660995538664314343558456, −5.58143296535932603870119505689, −4.45157843934770579182490154034, −3.71069828841462040582495135890, −1.69324692124496825225842676426, 2.74304496546748688791954417012, 3.14389428164062569822327045737, 3.96960201164235620015628570846, 5.77572832268506610733996780028, 7.22473695949368384010770269419, 8.019437756044724587634642769004, 8.799983710160092574563367789711, 10.17895581084310262789467195375, 11.10796885932589719971089371856, 11.71461336566885942551085495178

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