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Properties

Label 2-287-41.9-c1-0-9
Degree $2$
Conductor $287$
Sign $0.858 + 0.512i$
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  − 1.72i·2-s + (2.12 + 2.12i)3-s − 0.976·4-s − 0.471i·5-s + (3.66 − 3.66i)6-s + (0.707 + 0.707i)7-s − 1.76i·8-s + 6.04i·9-s − 0.812·10-s + (2.00 + 2.00i)11-s + (−2.07 − 2.07i)12-s + (−3.23 − 3.23i)13-s + (1.21 − 1.21i)14-s + (1.00 − 1.00i)15-s − 4.99·16-s + (−5.64 + 5.64i)17-s + ⋯
L(s)  = 1  − 1.21i·2-s + (1.22 + 1.22i)3-s − 0.488·4-s − 0.210i·5-s + (1.49 − 1.49i)6-s + (0.267 + 0.267i)7-s − 0.624i·8-s + 2.01i·9-s − 0.257·10-s + (0.603 + 0.603i)11-s + (−0.599 − 0.599i)12-s + (−0.897 − 0.897i)13-s + (0.326 − 0.326i)14-s + (0.258 − 0.258i)15-s − 1.24·16-s + (−1.37 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.858 + 0.512i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.858 + 0.512i$
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: \((2,\ 287,\ (\ :1/2),\ 0.858 + 0.512i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.86796 - 0.514617i\)
\(L(\frac12)\) \(\approx\) \(1.86796 - 0.514617i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.37 + 0.573i)T \)
good2 \( 1 + 1.72iT - 2T^{2} \)
3 \( 1 + (-2.12 - 2.12i)T + 3iT^{2} \)
5 \( 1 + 0.471iT - 5T^{2} \)
11 \( 1 + (-2.00 - 2.00i)T + 11iT^{2} \)
13 \( 1 + (3.23 + 3.23i)T + 13iT^{2} \)
17 \( 1 + (5.64 - 5.64i)T - 17iT^{2} \)
19 \( 1 + (-5.47 + 5.47i)T - 19iT^{2} \)
23 \( 1 - 2.83T + 23T^{2} \)
29 \( 1 + (6.45 + 6.45i)T + 29iT^{2} \)
31 \( 1 + 4.58T + 31T^{2} \)
37 \( 1 - 1.64T + 37T^{2} \)
43 \( 1 + 2.27iT - 43T^{2} \)
47 \( 1 + (6.12 - 6.12i)T - 47iT^{2} \)
53 \( 1 + (3.79 + 3.79i)T + 53iT^{2} \)
59 \( 1 - 1.71T + 59T^{2} \)
61 \( 1 + 3.67iT - 61T^{2} \)
67 \( 1 + (0.937 - 0.937i)T - 67iT^{2} \)
71 \( 1 + (-6.26 - 6.26i)T + 71iT^{2} \)
73 \( 1 - 14.7iT - 73T^{2} \)
79 \( 1 + (-2.09 - 2.09i)T + 79iT^{2} \)
83 \( 1 + 0.365T + 83T^{2} \)
89 \( 1 + (-2.42 - 2.42i)T + 89iT^{2} \)
97 \( 1 + (-1.49 + 1.49i)T - 97iT^{2} \)
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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

−11.41666760423222758681103631833, −10.78057260873958364387431050179, −9.734039622422289760423587597526, −9.331570168756094197685461250211, −8.383468368688500890787598182924, −7.07891766317995396397606548315, −5.02255968696196244314186701889, −4.13427488376331166034714998963, −3.07093156313239469689645461269, −2.07690970055220710185650945270, 1.85127467886592683303771847526, 3.23594615793470053235895387549, 5.02938915625666961663148881457, 6.54584602169730046968967711188, 7.13006679225847701572482243804, 7.67191743207560230943360740112, 8.833449988603086123527717020267, 9.295288603896675504511749311189, 11.22329385631073187649335791759, 12.01259705514982271855185089509

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