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Properties

Label 2-287-287.286-c2-0-22
Degree $2$
Conductor $287$
Sign $-0.0585 - 0.998i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.31·2-s + 5.31·3-s + 1.34·4-s + 9.29i·5-s − 12.2·6-s + (2.62 + 6.48i)7-s + 6.13·8-s + 19.2·9-s − 21.4i·10-s − 4.30i·11-s + 7.16·12-s − 4.06·13-s + (−6.06 − 15.0i)14-s + 49.4i·15-s − 19.5·16-s − 19.2·17-s + ⋯
L(s)  = 1  − 1.15·2-s + 1.77·3-s + 0.336·4-s + 1.85i·5-s − 2.04·6-s + (0.374 + 0.927i)7-s + 0.766·8-s + 2.14·9-s − 2.14i·10-s − 0.391i·11-s + 0.597·12-s − 0.313·13-s + (−0.433 − 1.07i)14-s + 3.29i·15-s − 1.22·16-s − 1.13·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0585 - 0.998i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.0585 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0585 - 0.998i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ -0.0585 - 0.998i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.04670 + 1.10983i\)
\(L(\frac12)\) \(\approx\) \(1.04670 + 1.10983i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-2.62 - 6.48i)T \)
41 \( 1 + (-38.8 - 13.1i)T \)
good2 \( 1 + 2.31T + 4T^{2} \)
3 \( 1 - 5.31T + 9T^{2} \)
5 \( 1 - 9.29iT - 25T^{2} \)
11 \( 1 + 4.30iT - 121T^{2} \)
13 \( 1 + 4.06T + 169T^{2} \)
17 \( 1 + 19.2T + 289T^{2} \)
19 \( 1 + 13.2T + 361T^{2} \)
23 \( 1 - 23.0T + 529T^{2} \)
29 \( 1 + 41.6iT - 841T^{2} \)
31 \( 1 - 19.3iT - 961T^{2} \)
37 \( 1 - 45.0T + 1.36e3T^{2} \)
43 \( 1 - 23.7T + 1.84e3T^{2} \)
47 \( 1 - 31.6T + 2.20e3T^{2} \)
53 \( 1 - 14.1iT - 2.80e3T^{2} \)
59 \( 1 + 102. iT - 3.48e3T^{2} \)
61 \( 1 - 37.3iT - 3.72e3T^{2} \)
67 \( 1 - 86.4iT - 4.48e3T^{2} \)
71 \( 1 + 20.8iT - 5.04e3T^{2} \)
73 \( 1 + 36.1iT - 5.32e3T^{2} \)
79 \( 1 + 73.3iT - 6.24e3T^{2} \)
83 \( 1 - 124. iT - 6.88e3T^{2} \)
89 \( 1 - 31.9T + 7.92e3T^{2} \)
97 \( 1 - 59.8T + 9.40e3T^{2} \)
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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

−11.33611633692223571121618776861, −10.63423378573561878275724819918, −9.640711220352485278856261363010, −8.996970952754237166202707209248, −8.128173783360566993657951923187, −7.44168430555443324372643705328, −6.45445467477245490443491478291, −4.21219523573094531067583661259, −2.75829417535951324789937335465, −2.19055568318628102761654830604, 0.956436155499423538340752200714, 2.05276290270671580503655936324, 4.17945021992438945555560570058, 4.68212489759414730568515197324, 7.21686404001750459878173513951, 7.87971561773725887013750642951, 8.762860240722128859702811078708, 9.056182521585064239790922290873, 9.860180034910552857284109444581, 10.95957188547204572877064439430

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