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Properties

Label 2-17e2-17.13-c1-0-6
Degree $2$
Conductor $289$
Sign $0.992 + 0.122i$
Analytic cond. $2.30767$
Root an. cond. $1.51910$
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  − 1.30i·2-s + (1.62 + 1.62i)3-s + 0.302·4-s + (0.921 + 0.921i)5-s + (2.12 − 2.12i)6-s + (−2.33 + 2.33i)7-s − 3i·8-s + 2.30i·9-s + (1.20 − 1.20i)10-s + (2.12 − 2.12i)11-s + (0.493 + 0.493i)12-s − 0.302·13-s + (3.04 + 3.04i)14-s + 3i·15-s − 3.30·16-s + ⋯
L(s)  = 1  − 0.921i·2-s + (0.940 + 0.940i)3-s + 0.151·4-s + (0.411 + 0.411i)5-s + (0.866 − 0.866i)6-s + (−0.882 + 0.882i)7-s − 1.06i·8-s + 0.767i·9-s + (0.379 − 0.379i)10-s + (0.639 − 0.639i)11-s + (0.142 + 0.142i)12-s − 0.0839·13-s + (0.813 + 0.813i)14-s + 0.774i·15-s − 0.825·16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.122i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(289\)    =    \(17^{2}\)
Sign: $0.992 + 0.122i$
Analytic conductor: \(2.30767\)
Root analytic conductor: \(1.51910\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{289} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 289,\ (\ :1/2),\ 0.992 + 0.122i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.88307 - 0.115472i\)
\(L(\frac12)\) \(\approx\) \(1.88307 - 0.115472i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad17 \( 1 \)
good2 \( 1 + 1.30iT - 2T^{2} \)
3 \( 1 + (-1.62 - 1.62i)T + 3iT^{2} \)
5 \( 1 + (-0.921 - 0.921i)T + 5iT^{2} \)
7 \( 1 + (2.33 - 2.33i)T - 7iT^{2} \)
11 \( 1 + (-2.12 + 2.12i)T - 11iT^{2} \)
13 \( 1 + 0.302T + 13T^{2} \)
19 \( 1 - 4.90iT - 19T^{2} \)
23 \( 1 + (0.921 - 0.921i)T - 23iT^{2} \)
29 \( 1 + (-0.642 - 0.642i)T + 29iT^{2} \)
31 \( 1 + (2.54 + 2.54i)T + 31iT^{2} \)
37 \( 1 + (4.67 + 4.67i)T + 37iT^{2} \)
41 \( 1 + (4.24 - 4.24i)T - 41iT^{2} \)
43 \( 1 + 9.60iT - 43T^{2} \)
47 \( 1 + 3T + 47T^{2} \)
53 \( 1 + 12.9iT - 53T^{2} \)
59 \( 1 - 6iT - 59T^{2} \)
61 \( 1 + (9.06 - 9.06i)T - 61iT^{2} \)
67 \( 1 - 5.39T + 67T^{2} \)
71 \( 1 + (-7.92 - 7.92i)T + 71iT^{2} \)
73 \( 1 + (5.37 + 5.37i)T + 73iT^{2} \)
79 \( 1 + (2.27 - 2.27i)T - 79iT^{2} \)
83 \( 1 - 15.5iT - 83T^{2} \)
89 \( 1 - 11.2T + 89T^{2} \)
97 \( 1 + (-0.0648 - 0.0648i)T + 97iT^{2} \)
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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

−11.78802453005990178294093082283, −10.60334164642735213844221684020, −9.922354280058155102533621417307, −9.311010078298319672236602435290, −8.407043911434247468739550648063, −6.75266029334048955007203651483, −5.79884356927884396955708442692, −3.87617449333693858079246336362, −3.21314619785641655841508629868, −2.18468268137448605247119320776, 1.70770596054140082785576484408, 3.12995396188710749163082519417, 4.85735284279872985974148795724, 6.39227467254521559443840603805, 6.98214834633392871323506230210, 7.67537825637459254488710396692, 8.765930147627795576816947821361, 9.561945108008296987191611980699, 10.83673262402822166783482722378, 12.14501792083512062319865468092

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