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Properties

Label 2-287-41.32-c1-0-19
Degree $2$
Conductor $287$
Sign $-0.603 - 0.797i$
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  − 2.74i·2-s + (0.521 − 0.521i)3-s − 5.51·4-s − 4.13i·5-s + (−1.43 − 1.43i)6-s + (−0.707 + 0.707i)7-s + 9.65i·8-s + 2.45i·9-s − 11.3·10-s + (1.67 − 1.67i)11-s + (−2.88 + 2.88i)12-s + (2.08 − 2.08i)13-s + (1.93 + 1.93i)14-s + (−2.15 − 2.15i)15-s + 15.4·16-s + (0.364 + 0.364i)17-s + ⋯
L(s)  = 1  − 1.93i·2-s + (0.301 − 0.301i)3-s − 2.75·4-s − 1.84i·5-s + (−0.584 − 0.584i)6-s + (−0.267 + 0.267i)7-s + 3.41i·8-s + 0.818i·9-s − 3.58·10-s + (0.506 − 0.506i)11-s + (−0.831 + 0.831i)12-s + (0.576 − 0.576i)13-s + (0.518 + 0.518i)14-s + (−0.557 − 0.557i)15-s + 3.85·16-s + (0.0884 + 0.0884i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.603 - 0.797i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.603 - 0.797i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (155, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.603 - 0.797i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.477353 + 0.960514i\)
\(L(\frac12)\) \(\approx\) \(0.477353 + 0.960514i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.27 + 6.27i)T \)
good2 \( 1 + 2.74iT - 2T^{2} \)
3 \( 1 + (-0.521 + 0.521i)T - 3iT^{2} \)
5 \( 1 + 4.13iT - 5T^{2} \)
11 \( 1 + (-1.67 + 1.67i)T - 11iT^{2} \)
13 \( 1 + (-2.08 + 2.08i)T - 13iT^{2} \)
17 \( 1 + (-0.364 - 0.364i)T + 17iT^{2} \)
19 \( 1 + (2.98 + 2.98i)T + 19iT^{2} \)
23 \( 1 - 3.81T + 23T^{2} \)
29 \( 1 + (3.49 - 3.49i)T - 29iT^{2} \)
31 \( 1 - 0.761T + 31T^{2} \)
37 \( 1 - 1.39T + 37T^{2} \)
43 \( 1 + 9.79iT - 43T^{2} \)
47 \( 1 + (-3.93 - 3.93i)T + 47iT^{2} \)
53 \( 1 + (7.31 - 7.31i)T - 53iT^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 + 6.57iT - 61T^{2} \)
67 \( 1 + (-3.88 - 3.88i)T + 67iT^{2} \)
71 \( 1 + (-9.41 + 9.41i)T - 71iT^{2} \)
73 \( 1 + 3.21iT - 73T^{2} \)
79 \( 1 + (3.26 - 3.26i)T - 79iT^{2} \)
83 \( 1 - 4.65T + 83T^{2} \)
89 \( 1 + (-2.35 + 2.35i)T - 89iT^{2} \)
97 \( 1 + (5.06 + 5.06i)T + 97iT^{2} \)
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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

−11.27677838695464364747463228710, −10.48145168433195577798594559273, −9.114521829260268211883877796993, −8.892460598387870860861765166797, −8.027433738841495488605915366692, −5.52377372746260028914954631677, −4.72848296773009152868794801971, −3.56478883315412399610591164804, −2.06580610588779539121390144323, −0.853586746109468741742562001790, 3.43934601779743819420230300580, 4.24906860614130489813627133544, 6.06724349873351842727472146588, 6.58791314738460952783360046102, 7.24320745395763380581539253292, 8.333695266936010033091877109482, 9.507286831040451934089017569850, 10.01041190881201764503101114640, 11.36520538598803969897644922049, 12.86318757086089689859506718751

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