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Properties

Label 2-287-41.40-c1-0-10
Degree $2$
Conductor $287$
Sign $-0.360 + 0.932i$
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  − 1.82·2-s − 1.19i·3-s + 1.31·4-s − 0.937·5-s + 2.17i·6-s + i·7-s + 1.25·8-s + 1.57·9-s + 1.70·10-s − 2.33i·11-s − 1.56i·12-s − 2.30i·13-s − 1.82i·14-s + 1.12i·15-s − 4.90·16-s − 4.88i·17-s + ⋯
L(s)  = 1  − 1.28·2-s − 0.690i·3-s + 0.656·4-s − 0.419·5-s + 0.888i·6-s + 0.377i·7-s + 0.442·8-s + 0.523·9-s + 0.539·10-s − 0.703i·11-s − 0.453i·12-s − 0.640i·13-s − 0.486i·14-s + 0.289i·15-s − 1.22·16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.360 + 0.932i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (204, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.360 + 0.932i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.286648 - 0.418061i\)
\(L(\frac12)\) \(\approx\) \(0.286648 - 0.418061i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 - iT \)
41 \( 1 + (5.97 + 2.30i)T \)
good2 \( 1 + 1.82T + 2T^{2} \)
3 \( 1 + 1.19iT - 3T^{2} \)
5 \( 1 + 0.937T + 5T^{2} \)
11 \( 1 + 2.33iT - 11T^{2} \)
13 \( 1 + 2.30iT - 13T^{2} \)
17 \( 1 + 4.88iT - 17T^{2} \)
19 \( 1 - 2.27iT - 19T^{2} \)
23 \( 1 + 7.58T + 23T^{2} \)
29 \( 1 + 9.66iT - 29T^{2} \)
31 \( 1 - 8.30T + 31T^{2} \)
37 \( 1 + 2.12T + 37T^{2} \)
43 \( 1 - 6.60T + 43T^{2} \)
47 \( 1 + 13.3iT - 47T^{2} \)
53 \( 1 - 9.55iT - 53T^{2} \)
59 \( 1 - 13.2T + 59T^{2} \)
61 \( 1 + 1.79T + 61T^{2} \)
67 \( 1 + 1.16iT - 67T^{2} \)
71 \( 1 - 8.32iT - 71T^{2} \)
73 \( 1 + 1.14T + 73T^{2} \)
79 \( 1 - 12.4iT - 79T^{2} \)
83 \( 1 + 0.290T + 83T^{2} \)
89 \( 1 - 0.0713iT - 89T^{2} \)
97 \( 1 + 14.6iT - 97T^{2} \)
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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

−11.60697272637972177407184863656, −10.23130870595485741185856790865, −9.730368375641303109708159330110, −8.363680181431530522063345139910, −7.963583737953772682493385293881, −7.01638630797191259951768266256, −5.79896533046254238393805343486, −4.14111064048756927960026776729, −2.25885845374327437975513660765, −0.60575748161309596718425625909, 1.68772725939601050888995456510, 3.92271515481365369067871602161, 4.69523194308582182347373601703, 6.57953900067918025345528110014, 7.54397549047732936424347901731, 8.402893608146835411018160650626, 9.418628014255067547670437497681, 10.12165902893108359257037386998, 10.70233237013220270989563069963, 11.79258451341909864227121199950

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